ResultsonSolutionsforSeveral -Painleve´Difference ...downloads.hindawi.com/journals/jmath/2020/3781942.pdf · (iii)If ξ 0 and Y(z) −Ahas ﬁnite zeros, then Ais a solutionof3z2

Research ArticleResults on Solutions for Several q-Painleve DifferenceEquations concerning Rational Solutions Zeros and Poles

Bu Sheng Li1 Rui Ying2 Xiu Min Zheng 3 and Hong Yan Xu 4

1Department of Informatics and Engineering Jingdezhen Ceramic Institute Jingdezhen 333403 Jiangxi China2Basic Department Shangrao Preschool Education College Shangrao 334001 Jiangxi China3Institute of Mathematics and Information Science Jiangxi Normal University Nanchan 330022 Jiangxi China4School of Mathematics and Computer Science Shangrao Normal University Shangrao 334001 Jiangxi China

Correspondence should be addressed to Hong Yan Xu xhyhhh126com

Received 30 May 2020 Revised 24 July 2020 Accepted 3 August 2020 Published 1 September 2020

Academic Editor Basil K Papadopoulos

Copyright copy 2020 Bu Sheng Li et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this article we discuss the problem about the properties on solutions for several types of q-difference equations and obtain someresults on the exceptional values of transcendental meromorphic solutions f(z) with zero order their q-differencesΔqf(z) f(qz) minus f(z) and divided differences Δqf(z)f(z) In addition we also investigated the condition on the existence ofrational solution for a class of q-difference equations Our theorems are some extensions and supplement to those results given byLiu and Zhang and Qi and Yang

1 Introduction and Main Results

All the time Painleve equations have attracted much interestdue to the reduction of solution equations which aresolvable by inverse scattering transformations and theyoften occur in many physical situations plasma physicsstatistical mechanics and nonlinear waves e study ofPainleve equations has spanned more than one hundredyears (see [1ndash3])

Around 2006 Halburd and Korhonen [4 5] and Ron-kainen [6] used Nevanlinna theory to discuss the followingequations

f(z + 1) + f(z minus 1) R(z f)

P(z f)

Q(z f)

f(z + 1)f(z minus 1) R(z f)

P(z f)

Q(z f)

(1)

where R(z f) is rational in f and meromorphic in z re-spectively and they singled out the following differenceequations

f(z + 1) + f(z minus 1) az + b

f(z)+ c (2)

f(z + 1) + f(z minus 1) (az + b)f(z) + c

1 minus f(z)2 (3)

f(z + 1)f(z minus 1) η(z)f(z)

2minus λ(z)f(z) + μ(z)

(f(z) minus 1)(f(z) minus υ(z)) (4)

f(z + 1)f(z minus 1) η(z)f(z)

2minus λ(z)f(z)

f(z) minus 1 (5)

f(z + 1)f(z minus 1) η(z)(f(z) minus λ(z))

(f(z) minus 1) (6)

f(z + 1)f(z minus 1) h(z)f(z)m

(7)

HindawiJournal of MathematicsVolume 2020 Article ID 3781942 10 pageshttpsdoiorg10115520203781942

where η(z) λ(z) and υ(z) satisfy some conditions In theseequations equation (2) is called as the difference Painleve Iequation equation (3) is called as the difference Painleve IIequation and the last four equations are called as the dif-ference Painleve III equations

In the last decade or so there were a lot of papers fo-cusing on the properties of solutions for difference PainleveIndashIV equations (see [7ndash11]) For example Chen and Shon[12] in 2010 considered the difference Painleve I equation (3)and obtained the following theorem

Theorem 1 (see [12] eorem 4) Let a b c be constantswhere a b are not both equal to zero 2en the followingholds

(i) If ane 0 then (3) has no rational solution(ii) If a 0 and bne 0 then (3) has a nonzero constant

solution w(z) A where A satisfies 2A2 minus cAminus

b 0

e other rational solution w(z) satisfies w(z) (P(z)Q(z)) + A where P(z) and Q(z) are relatively primepolynomials and satisfy degPlt degQ

In 2013 and 2018 Zhang and Yi [11] and Du et al [13]studied the difference Painleve III equations with the con-stant coefficients and obtained the result as follows

Theorem 2 (see [11 13]) Iff is a transcendental finite-ordermeromorphic solution of

f(z + 1)f(z minus 1)(f(z) minus 1)2

f(z)2

minus λf(z) + μ (8)

where λ and μ are constants then the following holds

(i) τ(f) σ(f)(ii) If λμne 0 then λ(f) σ(f)(iii) For any η isin C 0 τ(f(z + η)) σ(f)(iv) λ(1Δf) λ(1(Δff)) σ(f)

Ramani et al [14] in 2003 investigated the existence oftranscendental solution of equation

(f(z + 1) + f(z))(f(z) + f(z minus 1)) R(z f)

P(z f)

Q(z f)

(9)

which is called as difference Painleve IV equations andobtained the result as follows

Theorem 3 (see [14]) If the second-order difference equation(9) admits a nonrational meromorphic solution of finiteorder then degzPle 4 and degzQle 2

Of late many mathematicians paid considerable atten-tion to the value distribution of solutions for complexq-difference equations which are formed by replacing theq-difference f(qz) q isin C 0 1 with f(z + c) of mero-morphic function in some expressions concerning complexdifference equations by utilizing the logarithmic derivative

lemma on q-difference operators given by Barnett et al [15]in 2007 (see [16ndash26]) For example Qi and Yang [27]considered the following equation

f(qz) + fz

q1113888 1113889

az + b

f(z)+ c (10)

which can be seen as q-difference analogues of (2) andobtained the result as follows

Theorem 4 (see [27] eorem 1) Let f(z) be a tran-scendental meromorphic solution with zero order of equation(10) and a b c be three constants such that a b cannot vanishsimultaneously 2en the following holds

(i) f(z) has infinitely many poles(ii) If ane 0 and any d isin C then f(z) minus d has infinitely

many zeros(iii) If a 0 and f(z) takes a finite value A finitely often

then A is a solution of 2z2 minus cz minus b 0

In 2018 Liu and Zhang [28] further investigated thefollowing equation

Y(ωz) + Y(z) + Yz

ω1113874 1113875

ξz + o

Y(z)+ ] (11)

and obtained the result as follows

Theorem 5 (see [28] eorem 1) Let Y(z) be a tran-scendental meromorphic solution with zero order of (11) andξ o ] be three constants such that ξ o cannot vanish si-multaneously 2en the following holds

(i) Y(z) has infinitely many poles(ii) For any finite value B if ξ 0 then Y(z) minus B has

infinitely many zeros(iii) If ξ 0 and Y(z) minus A has finite zeros then A is a

solution of 3z2 minus o minus ]z 0

Motivated by the idea [27 28] a natural question is whatis the result if we give q-difference analogues of (9) For thisquestion our main aim of this article is further to investigatesome properties of meromorphic solutions for someq-Painleve difference IV equations It seems that this topichas never been treated before

In what follows it should be assumed that the readers arefamiliar with the fundamental results and the standardnotations in the theory of Nevanlinna value distribution (seeHayman [29] Yang [30] and Yi and Yang [31]) Let f be ameromorphic function and we denote σ(f) λ(f) andλ(1f) to be the order the exponent of convergence of zerosand the exponent of convergence of poles of f(z) re-spectively and denote τ(f) to be the exponent of conver-gence of fixed points of f(z) which is defined by

τ(f) lim supr⟶+infin

logN(r 1(f(z) minus z))

log r (12)

2 Journal of Mathematics

In addition we use S(r f) denotes any quantity satis-fying S(r f) o(T(r f)) for all r on a set F of logarithmicdensity 1 and the logarithmic density of a set F is defined by

lim supr⟶infin

1log r

1113946[1r]capF

1tdt (13)

Now our main results are listed as follows

Theorem 6 Let R(z) A(z)B(z) be an irreducible ra-tional function and let q(ne 0) isin C and |q|ne 1 and

[f(qz) + f(z)] f(z) + fz

q1113888 11138891113890 1113891 R(z)

A(z)

B(z)

(14)

where A(z) B(z) are polynomials with degzA(z) a anddegzB(z) b

(i) Suppose that age b and a minus b are even numbers orzero If equation (14) has an irreducible rationalsolution f(z) μ(z)](z) where μ(z) ](z) arepolynomials with degz μ(z) μ and degz ](z) ]then the following holds

μ minus ] a minus b

2 (15)

(ii) Suppose that bge a and b minus a are even numbers or zeroIf equation (14) has an irreducible rational solutionf(z) μ(z)](z) where μ(z) ](z) are polynomialswith degz μ(z) μ and degz ](z) ] then

] minus μ b minus a

2 (16)

(iii) If |a minus b| is an odd number then equation (14) hasno rational solution

Theorem 7 For q(ne 0) isin C and |q|ne 1 let f(z) be atranscendental meromorphic solution with zero order ofequation

[f(qz) + f(z)] f(z) + fz

q1113888 11138891113888 1113889 af(z)

2 (17)

where a(ne 0 4) is a constant Let Δqf f(qz) minus f(z) 2enthe following holds

(i) Both f and Δqf have no nonzero finite Nevanlinnaexceptional value

(ii) If q + (1q)ne a minus 2 then f(ηz) and Δqf(ηz) haveinfinitely many fixed points andτ(f(ηz)) τ(Δqf(ηz)) σ(f) for any nonzeroconstant η

Theorem 8 For q(ne 0) isin C and |q|ne 1 and let f(z) be atranscendental meromorphic solution with zero order ofequation

[f(qz) + f(z)] f(z) + fz

q1113888 11138891113890 1113891 d(z)f(z) (18)

where d(z) is a nonconstant rational function satisfying thatd(qz)d(z) is not a constant 2en the following holds

(i) Both f and Δqff have no Nevanlinna exceptionalvalue

(ii) Δqf has infinitely many poles and zeros andλ(1Δqf) λ(Δqf) σ(f)

(iii) Δqff has infinitely many fixed points andτ(Δqff) σ(f)

2 Proof of Theorem 6

Proof assume that (14) has a rational solutionf(z) μ(z)v(z) and has poles z1 z2 zk en f(z)

can be represented in the following form

f(z) μ(z)

](z) 1113944

k

j1

cjλj

z minus zj1113872 1113873λj

+ middot middot middot +cj1

z minus zj1113872 1113873

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

+ a0 + a1z + middot middot middot + aszs

(19)

where cjλj(ne 0) cj1(j 1 2 k) and a0 a1 as

are constants zj(j 1 2 k) are poles of f(z) withmultiplicity λj respectively

(i) Suppose that agt b and a minus b are even numbers enin view of (14) and (19) it yields

μ(qz)

](qz)+μ(z)

](z)1113888 1113889

μ(z)

](z)+μ(zq)

](zq)1113888 1113889

A(z)

B(z) (20)

If degz μ(z) μlt ] degz](z) then for z⟶ infin itfollows

μ(qz)

](qz)⟶ 0

μ(z)

](z)⟶ 0

μ(zq)

](zq)⟶ 0

(21)

However A(z)B(z)⟶ infin as z⟶ infin thus from(20) we can get a contradiction easilyIf μ ] then let z⟶ infin and it leads to

μ(qz)

](qz)⟶ α

μ(z)

](z)⟶ α

μ(zq)

](zq)⟶ α

(22)

Journal of Mathematics 3

where α is a nonzero constant us let z⟶ +infin inview of (20) we also get a contradiction So it followsμgt ] us assume that as ne 0 (sge 1) where s μ minus ]As z⟶ infin it yields

f(z) aszs(1 + o(1))

f(qz) asqsz

s(1 + o(1))

fz

q1113888 1113889 asq

minus sz

s(1 + o(1))

A(z)

B(z) βz

aminus b(1 + o(1))

(23)

where β(ne 0) is a constant and it follows now in viewof (20) that

2 + qs

+ qminus s

( 1113857a2s z

2s(1 + o(1)) βz

aminus b(1 + o(1))

(24)

as z⟶ infin Since |q|ne 1 then qs + 2 + qminus s ne 0 Henceit follows from (24) that

μ minus ] s

a minus b

2

(25)

Next assume that a b As z⟶ infin it followsA(z)

B(z) β(1 + o(1)) (26)

where β(ne 0) is a constant If μlt ] then by using thesame argument as above we get a contradiction Ifμge ] then we assume that as ne 0 (sge 1) By using thesame argument as above we conclude

qs

+ 2 + qminus s

1113858 1113859a2s z

2s β(1 + o(1)) (27)

as z⟶ infin us if μgt ] then in view of (27) we canget a contradiction if μ ] then we have

μ minus ] 0

a minus b

2

(28)

(ii) Suppose that bgt a and b minus a are even numbersenin view of (14) and (19) we get (20)If μgt ] then for z⟶ infin it leads to

μ(qz)

](qz)⟶ infin

μ(z)

](z)⟶ infin

μ(zq)

](zq)⟶ infin

(29)

However A(z)B(z)⟶ 0 as z⟶ infin thus from(20) we can get a contradiction easilyIf μ ] then let z⟶ infin it follows

μ(qz)

](qz)⟶ α

μ(z)

](z)⟶ α

μ(zq)

](zq)⟶ α

(30)

where α is a nonzero constant us let z⟶ +infin inview of (20) we also get a contradictionus μlt ] Werewrite (14) as the following form

B(z) μ(qz)μ(z)]z

q1113888 1113889](z) + μ(qz)μ

z

q1113888 1113889](z)

21113890

+ μ(z)2](qz)]

z

q1113888 1113889 + μ(z)μ

z

q1113888 1113889](qz)](z)1113891

A(z)](z)2](qz)]

z

q1113888 1113889

(31)

Denote

A(z) ξaza

+ middot middot middot

B(z) δbzb


μ(z) cμzμ


](z) ζ]z]


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(32)

where age 1 bge 0 and μge 0 ]ge 1 are all nonnegativeintegers us in view of (31) and (32) we can deduce

qμminus ]

+ 2 + q]minus μ

( 1113857δbc2μζ

2]z

2(μ+])+b+ middot middot middot ξaζ

4]z

4]+a

(33)

Since |q|ne 1 then qμminus ] + 2 + q]minus μ ne 0 us by com-bining with this and (33) we have

2(μ + ]) + b 4] + a that is ] minus μ b minus a

2 (34)

and ζ2]c2μ (δbξa)(qμminus ] + 2 + q]minus μ)

(iii) If agt b then |a minus b| a minus b is an odd numberAssume that f(z) μ(z)](z) is a rational solutionof (14) In view of the conclusion ofeorem 6 (i) itfollows μ minus ] (a minus b)2 is means a contradic-tion with the assumption that a minus b is an oddnumber us (14) has no rational solution

If alt b then |a minus b| b minus a is an odd number Similar tothe above argument we also conclude that (14) has norational solution


erefore this completes the proof of eorem 6


We first introduce some notations and some basic resultsabout Nevanlinna theory which can be used in Section 3 andSection 4 Let f be a meromorphic function in C theNevanlinna characteristic T(r f) which encodes infor-mation about the distribution of values of f on the disk|z|le r is defined by

T(r f) m(r f) + N(r f) (35)

e proximity function m(r f) is defined by

m(r f) m(rinfin f)

12π

11139462π

0log+

f reiθ1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868dθ

(36)

where log+x max 0 log x1113864 1113865 and

N(r f) N(rinfin f)

1113946r

0

n(t f) minus n(0 f)

tdt + n(0 f)log r

(37)

where n(r f) is the number of poles of f in the circle |z| rcounted according to multiplicities

Let a isin Ccup infin and the deficiency of a with respect tof(z) is defined by

δ(a f) lim infr⟶+infin

m(r (1f minus a))

T(r f)

1 minus lim supr⟶+infin

N(r (1f minus a))

T(r f)

(38)

If δ(a f)gt 0 then the complex number a is called theNevanlinna exceptional value And the order σ(f) theexponent of convergence of zeros λ(f) and the exponent ofconvergence of poles λ(1f) of f(z) are defined by

σ(f) lim supr⟶+infin

log+T(r f)

log r

λ(f) lim supr⟶+infin

log+N(r 1f)

log r

λ1f

1113888 1113889 lim supr⟶+infin

log+N(r f)

log r

(39)

Besides we also use some properties ofT(r f) m(r f) andN(r f) such as

T r1

f minus a1113888 1113889 T(r f) + O(1)

m r 1113945

p

j1fj

⎛⎝ ⎞⎠le 1113944

p

j1m r fj1113872 1113873 m r 1113944

p

j1fj

⎛⎝ ⎞⎠

le 1113944

p

j1N r fj1113872 1113873 + logp

N r 1113945

p

j1fj

⎛⎝ ⎞⎠le 1113944

p

j1N r fj1113872 1113873 N r 1113944

p

j1fj

⎛⎝ ⎞⎠le 1113944

p

j1N r fj1113872 1113873

T r 1113945

p

j1fj

⎛⎝ ⎞⎠le 1113944

p

j1T r fj1113872 1113873 T r 1113944

p

j1fj

⎛⎝ ⎞⎠

le 1113944

p

j1T r fj1113872 1113873 + logp

(40)

where fj(z)(j 1 2 p) are p meromorphic functionsand a isin C and require some lemmas as follows

Lemma 1 (see [15] eorem 2) Let f be a nonconstantzero-order meromorphic solution of P(z f) 0 whereP(z f) is a q-difference polynomial in f(z) If P(z a)≢ 0 forslowly moving target a(z) then

m r1

f minus a1113888 1113889 S(r f) (41)

where S(r f) denotes any quantity satisfyingS(r f) o(T(r f)) for all r on a set F of logarithmic density1

Remark 1 For q isin C 0 1 a polynomial in f(z) and fi-nitely many of its q-shifts f(qz) f(qnz) with mero-morphic coefficients in the sense that their Nevanlinnacharacteristic functions are o(T(r f)) on a set F of loga-rithmic density 1 and can be called as a q-difference poly-nomial of f

Lemma 2 (see [24] eorems 1 and 3) Let f(z) be anonconstant zero-order meromorphic function and q isin C 0 2en

T(r f(qz)) (1 + o(1))T(r f(z))

N(r f(qz)) (1 + o(1))N(r f(z))(42)

on a set of lower logarithmic density 1


2e proof of 2eorem 7 (i) suppose that f(z) is a tran-scendental meromorphic solution of equation (17) then inview of (17) let

P1(z f) ≔ [f(qz) + f(z)] f(z) + fz

q1113888 11138891113890 1113891 minus af(z)

2 equiv 0

(43)

For any given constant d isin C 0 and with a view ofane 4 it follows

P1(z d) 4d2

minus ad2 ne 0 (44)

In view of P1(z d)≢ 0 and by Lemma 1 we concludethat m(r (1f minus d)) S(r f) is leads to

N r1

f minus d1113888 1113889 T(r f) + S(r f) (45)

which implies δ(d f) 0 us f(z) has no nonzero finiteNevanlinna exceptional value

Since f(z) is of zero order and Δqf f(qz) minus f(z)then by Lemma 2 it follows T(rΔqf)le 2T(r f) + S(r f)which means that Δqf if of zero order In view of (17) itfollows

f q2z1113872 1113873 + f(qz)1113960 1113961[f(qz) + f(z)] af(qz)

2 (46)

With (17) subtraction it leads to

[f(qz) + f(z)] f q2z1113872 1113873 + f(qz) minus f(z) minus f

z

q1113888 11138891113890 1113891

a[f(qz) + f(z)][f(qz) minus f(z)]

(47)

From (17) we see that f(qz) + f(z) equiv 0 Otherwise itleads to f(z) equiv 0 a contradiction us the above equalitymeans

f q2z1113872 1113873 + f(qz) minus f(z) minus f

z

q1113888 1113889 a[f(qz) minus f(z)]

(48)

that is

Δqf(qz) +(2 minus a)Δqf(z) + Δqfz

q1113888 1113889 0 (49)

Denote

P2 zΔqf1113872 1113873 ≔ Δqf(qz) +(2 minus a)Δqf(z)

+ Δqfz

q1113888 1113889 equiv 0

(50)

For any given constant d isin C 0 then from (50) wehave P2(z d) (4 minus a)d Hence with a view of ane 4 itfollows P2(z d) equiv 0 us by Lemma 1 we havem(r 1Δqf minus d) S(rΔqf) and this leads to

N r1Δqf minus d

1113888 1113889 T rΔqf1113872 1113873 + S rΔqf1113872 1113873 (51)

which implies that δ(dΔqf) 0 us Δqf has no nonzerofinite Nevanlinna exceptional value

(ii) Replacing z by ηz in (17) we have

[f(qηz) + f(ηz)] f(ηz) + fηz

q1113888 11138891113890 1113891 af(ηz)

2

(52)

Let g1(z) f(ηz) it yields

g1(qz) + g1(z)1113858 1113859 g1(z) + g1z

q1113888 11138891113890 1113891 ag1(z)

2 (53)

Set

P3 z g1( 1113857 ≔ g1(qz) + g1(z)1113858 1113859

middot g1(z) + g1z

q1113888 11138891113890 1113891 minus ag1(z)

2 equiv 0

(54)

us it follows P3(z z) (((q + 1)2q) minus a)z2 and with aview of q + (1q)ne a minus 2 we have P3(z z) equiv 0 By applyingLemma 1 it yields m(r 1(g1(z) minus z)) S(r g1) us inview of Lemma 2 this leads to

N r1

f(ηz) minus z1113888 1113889 N r

1g1(z) minus z

1113888 1113889

T r g1(z)( 1113857 + S r g1( 1113857

T(r f(ηz)) + S(r f(ηz))

T(r f) + S(r f)

(55)

which implies that f(ηz) has infinitely many fixed pointsand τ(f(ηz)) σ(f)

In view of (48) set g2(z) Δqf(ηz) and

P4 z g2( 1113857 ≔ g2(qz) +(2 minus a)g2(z) + g2z

q1113888 1113889 equiv 0 (56)

then P4(z z) [q + (2 minus a) + (1q)]z Sinceq + (1q)ne a minus 2 then P4(z z) equiv 0 By applying Lemma 1we have m(r (1(g2(z) minus z))) S(r g2) us in view ofLemma 2 this leads to

N r1

Δqf(ηz) minus z1113888 1113889 N r

1g2(z) minus z

1113888 1113889

T r g2(z)( 1113857 + S r g1( 1113857

T rΔqf(ηz)1113872 1113873 + S(r f(ηz))

(57)

which implies that Δqf(ηz) has infinitely many fixed pointsand τ(Δqf(ηz)) σ(f)

erefore the proof of eorem 7 is completed



Lemma 3 (see [15] eorem 1) Let f(z) be a nonconstantzero-order meromorphic function and q isin C 0 2en

m rf(qz)

f(z)1113888 1113889 S(r f) (58)

Lemma 4 (see [17] eorem 25) Let f be a transcendentalmeromorphic solution of order zero of a q-difference equationof the form

Uq(z f)Pq(z f) Qq(z f) (59)

where Uq(z f) Pq(z f) and Qq(z f) are q-differencepolynomials such that the total degree degUq(z f) n inf(z) and its q-shifts whereas degQq(z f)le n Moreover weassume that Uq(z f) contains just one term of maximal totaldegree in f(z) and its q-shifts 2en

m r Pq(z f)1113872 1113873 S(r f) (60)

Proof of 2eorem 8 (i) suppose that f(z) is a transcen-dental meromorphic solution of equation (18) We firstlyprove that Δqff has no Nevanlinna exceptional valueEquation (18) can be rewritten as

f(qz)

f(z)+ 11113888 1113889

f(zq)

f(z)+ 11113888 1113889

d(z)

f(z) (61)

Set g3(z) f(qz)f(z) In view of (61) and Lemma 2 itfollows

T(r f(z)) T rd(z)

f(z)1113888 1113889 + O(log r)leT r

f(qz)

f(z)1113888 1113889

+ T rf(zq)

f(z)1113888 1113889 + O(log r)

T r g3(z)( 1113857 + T r g3z

q1113888 11138891113888 1113889 + O(log r)

2T r g3( 1113857 + S r g3( 1113857

2T rΔqf

f1113888 1113889 + S(r f)

(62)

And with a view of T(rΔqff)le 2T(r f) + S(r f) wethus conclude that Δqff is transcendental and of orderzero and d(z) is small with respect to g3(z)

Replacing z by qz in (18) it follows

f q2z1113872 1113873 + f(qz)1113960 1113961[f(qz) + f(z)] d(qz)f(qz) (63)

By combining with (18) we have

f q2z1113872 1113873 + f(qz)

f(z) + f(zq)

d(qz)f(qz)

d(z)f(z) (64)

Since g3(z) f(qz)f(z) then it yields

f(qz) g3(z)f(z)

f q2z1113872 1113873 g3(qz)f(qz)

g3(z)g3(qz)f(z)

fz

q1113888 1113889

f(z)

g3(zq)

(65)

Substituting (65) into (64) we obtain

g3(z)g3(qz)f(z) + g3(z)f(z)

f(z) + f(z)g3(zq)( 1113857

d(qz)

d(z)g3(z) (66)

that is

g3z

q1113888 1113889 g3(qz) + 1( 1113857

d(qz)

d(z)g

z

q1113888 1113889 + 11113888 1113889 (67)

By applying Lemma 4 for (67) it followsm(r g3(zq)) S(r g3) is leads to

N r g3z

q1113888 11138891113888 1113889 T r g3( 1113857 + S r g3( 1113857 (68)

us in view of Lemma 2 it yields

(1 + o(1))N r g3( 1113857 N r g3z

q1113888 11138891113888 1113889

T r g3z

q1113888 11138891113888 1113889 + S r g3( 1113857

(1 + o(1))T r g3( 1113857 + S r g3( 1113857

(69)

is shows

N rΔqf

f1113888 1113889 N r g3( 1113857

T r g3( 1113857 + S r g3( 1113857

T rΔqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(70)

which implies δ(infinΔqff) 0Set

P5 z g3( 1113857 ≔ g3z

q1113888 1113889 g3(qz) + 1( 1113857

minusd(qz)

d(z)g

z

q1113888 1113889 + 11113888 1113889 equiv 0

(71)

For any constant ϱ isin C minus2 we have


P5(z ϱ + 1) (ϱ + 1)(ϱ + 1 + 1) minusd(qz)

d(z)(ϱ + 1 + 1)

(ϱ + 2) ϱ + 1 minusd(qz)

d(z)1113888 1113889

(72)

Since d(qz)d(z) is not a constant then P5(z ϱ + 1)≢ 0By Lemma 1 it follows

m r1

Δqff1113872 1113873 minus ϱ⎛⎝ ⎞⎠ m r

1g3 minus ϱ minus 1

1113888 1113889

S r g3( 1113857 S rΔqf

f1113888 1113889

(73)

is means

N r1

Δqff1113872 1113873 minus ϱ⎛⎝ ⎞⎠ T r

Δqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(74)

which implies that δ(ϱΔqff) 0 for any constantϱ isin C minus2

In view of (18) and Lemma 3 we have

m r1

f(qz) + f(z)1113888 1113889 m r

f(z) + f(zq)

d(z)f(z)1113888 1113889

lem r1

d(z)1113888 1113889 + m r

f(z) + f(zq)

f(z)1113888 1113889

+ O(1)

le S(r f)

(75)

us we can conclude from (18) (75) and Lemma 2 that

N r1

f(qz) + f(z)1113888 1113889 T r

1f(qz) + f(z)

1113888 1113889 + S(r f)

T r1

d(z)

f(z) + f(zq)

f(z)1113888 1113889 + S(r f)

T rf(zq)

f(z)1113888 1113889 + O(log r) + S(r f)

T rf(qz)

f(z)1113888 1113889 + S(r f)

T rΔqf

f1113888 1113889 + S(r f)

(76)

On the contrary we can see that the zero of (Δqff) + 2is the zero of f(qz) + f(z) and the zero of f(qz) + f(z) isalso the zero of (Δqff) + 2 Indeed if z0 is a zero of(Δqff) + 2 that is ((f(qz0) minus f(z0))f(z0)) + 2 0

then it follows f(qz0) + f(z0) 0 and this shows that z0 isa zero of f(qz) + f(z) if z1 is a zero of f(qz) + f(z) thatis f(qz1) + f(z1) 0 then it follows (Δqf(qz1)f(z1))+

2 0 and this shows that z1 is a zero of (Δqff) + 2 usby combining with (76) we conclude

N r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ N r

1f(qz) + f(z)

1113888 1113889

T rΔqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(77)

is shows that δ(minus2Δqff) 0 thus by combiningwith δ(infinΔqff) 0 and δ(ϱΔqff) 0 for anyϱ isin C minus2 we have δ(ϱΔqff) 0 for any ϱ isin C SoΔqff has no Nevanlinna exceptional value

Next we prove that f(z) has no Nevanlinna exceptionalvalue

Firstly in view of (61) and Lemma 3 we have

m r1f

1113888 1113889lem r1

d(z)1113888 1113889 + m r

f(qz)

f(z)1113888 1113889 + m r

f(zq)

f(z)1113888 1113889

S(r f)

(78)

which implies

N r1f

1113888 1113889 T(r f) + S(r f) (79)

is means δ(0 f) 0Secondly in view of (18) we denote

P6(z f) ≔ [f(qz) + f(z)] f(z) + fz

q1113888 11138891113890 1113891 minus d(z)f(z) equiv 0

(80)

Since d(z) is a nonconstant function then for anyconstant ϱ isin C 0 it yields

P6(z ϱ) 4ϱ2 minus ϱ d(z)

ϱ(4ϱ minus d(z))≢ 0(81)

us from Lemma 1 we conclude m(r 1f minus ϱ)

S(r f) which implies

N r1

f minus ϱ1113888 1113889 T(r f) + S(r f) (82)

is means that δ(ϱ f) 0 for any constant ϱ isin C 0 Finally in view of (77) we have

m r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ S r

Δqf

f1113888 1113889

S(r f)

(83)

us from (18) and (83) it yields


m r f(z) + fz

q1113888 11138891113888 1113889 m r

d(z)f(z)

f(qz) + f(z)1113888 1113889

m r1

(f(qz)f(z)) + 11113888 1113889 + O(log r)

m r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ + O(log r)

S(r f)

(84)

Hence by the above equality and in view of (18) andLemma 3 it leads to

m(r f)lem r f(z) + fz

q1113888 11138891113888 1113889 + m(r f(qz) + f(z))

+ O(log r)

le 2m r f(z) + fz

q1113888 11138891113888 1113889 + m r

f(qz) + f(z)

f(z) + f(zq)1113888 1113889

+ O(log r)

S(r f)

(85)

which implies

N(r f) T(r f) + S(r f) (86)

Hence δ(infin f) 0 Together with δ(0 f) 0 andδ(ϱ f) 0 for any ϱ isin C 0 we obtain that f(z) has noNevanlinna exceptional value

(ii) Since f(qz) Δqf(z) + f(z) andf(zq) f(z) minus Δqf(zq) then by substitutingthese into (18) it follows

2f(z) + Δqf(z)1113960 1113961 2f(z) minus Δqfz

q1113888 11138891113890 1113891 d(z)f(z)

(87)

If z0 is a zero of f(z) and is not a pole of d(z) then inview of (87) we conclude that z0 is a zero of Δqf(z) ora zero of Δqf(zq) us it follows from Lemma 2 that

N r1f

1113888 1113889leN r1Δqf(z)

1113888 1113889 + N r1

Δqf(zq)1113888 1113889 + N(r d)

2N r1Δqf

1113888 1113889 + S(r f)

(88)

us by combining with (79) it yields that Δq f(z) hasinfinitely many zeros and λ(Δqf) σ(f)

In view of (85) and Lemma 3 we can deduce

m rΔqf1113872 1113873lem(r f) + m rΔqf

f1113888 1113889 S(r f)

S rΔqf1113872 1113873

(89)

which implies

N rΔqf1113872 1113873 T rΔqf1113872 1113873 + S rΔqf1113872 1113873

T r1Δqf

1113888 1113889 + S rΔqf1113872 1113873

(90)

us by combining with (88) and (79) we have thatΔqf(z) has infinitely many poles andλ(1Δqf) σ(f) is proves the conclusions ofeorem 8 (ii)

(iii) In view of (71) it follows

P5(z z + 1) z

q+ 11113888 1113889(qz + 1 + 1) minus

d(qz)

d(z)

z

q+ 1 + 11113888 1113889

(z + q)(qz + 2)

qminus

d(qz)

d(z)

z + 2q

q

(91)

Since d(z) is a nonconstant rational function then letz⟶ infin and we have d(qz)d(z)⟶ qκ whereκ degz d But for qne 0 it follows((z + q)(qz + 2)z + 2q)⟶ infin as z⟶ infin us we candeduce P5(z z + 1) equiv 0 By Lemma 1 we conclude that

m r1

Δqff1113872 1113873 minus z⎛⎝ ⎞⎠ m r

1g3(z) minus 1 minus z

1113888 1113889

S r g3( 1113857

S rΔqf

f1113888 1113889

(92)

is leads to

N r1

Δqff1113872 1113873 minus z⎛⎝ ⎞⎠ T r

Δqf

f1113888 1113889 + S r

Δqf

f1113888 1113889 (93)

which implies that Δqff has infinitely many fixed pointsand τ(Δqff) σ(f)


Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest


Authorsrsquo Contributions

H Y Xu and XM Zheng conceptualized the study H Y Xuand R Ying wrote the original draft H Y Xu B S Li andX M Zheng reviewed and edited the manuscript H Y Xuand B S Li obtained funding acquisition

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (no 11561033) Natural ScienceFoundation of Jiangxi Province in China (nos20181BAB201001 and 20151BAB201008) and Foundation ofEducation Department of Jiangxi (nos GJJ190876GJJ190895 and GJJ191042) of China

References

[1] A S Fokas B Grammaticos and A Ramani ldquoFrom con-tinuous to discrete Painleve equationsrdquo Journal of Mathe-matical Analysis and Applications vol 180 no 2 pp 342ndash3601993

[2] B Grammaticos F W Nijhoff and A Ramani ldquoDiscretePainleve equationsrdquo 2e Painleve Property CRM Series inMathematical Physics Springer New York NY USApp 413ndash516 1999

[3] P Painleve ldquoMemoire sur les equations differentielles dontlrsquointegrale generale est uniformerdquo Bulletin de la Societemathematique de France vol 28 pp 201ndash261 1900

[4] R G Halburd and R J Korhonen ldquoDifference analogue of thelemma on the logarithmic derivative with applications todifference equationsrdquo Journal of Mathematical Analysis andApplications vol 314 no 2 pp 477ndash487 2006

[5] R G Halburd and R J Korhonen ldquoFinite-order meromor-phic solutions and the discrete Painleve equationsrdquo Pro-ceedings of the London Mathematical Society vol 94 no 2pp 443ndash474 2007

[6] O Ronkainen ldquoMeromorphic solutions of difference Painleveequationsrdquo PhD thesis Department of Physics and Math-ematics University of Eastern Finland Finland 2010

[7] Z X Chen ldquoComplex differences and difference equationsrdquoMathematics Monograph Series p 29 Science Press BeijingChina 2014

[8] Z-T Wen ldquoFinite order solutions of difference equations anddifference Painleve equations IVrdquo Proceedings of the AmericanMathematical Society vol 144 no 10 pp 4247ndash4260 2016

[9] H Y Xu and J Tu ldquoExistence of rational solutions for q-difference Painleve equationsrdquo Electronic Journal of Differ-ential Equations vol 2020 no 14 pp 1ndash14 2020

[10] J Zhang ldquoSome results on difference Painleve IV equationsrdquoJournal of Difference Equations and Applications vol 22no 12 pp 1912ndash1929 2016

[11] J Zhang and H Yi ldquoProperties of meromorphic solutions ofPainleve III difference equationsrdquo Advances in DifferenceEquations vol 2013 no 1 p 256 2013

[12] Z-X Chen and K H Shon ldquoValue distribution of mero-morphic solutions of certain difference Painleve equationsrdquoJournal of Mathematical Analysis and Applications vol 364no 2 pp 556ndash566 2010

[13] Y F Du M F Chen Z S Gao and M Zhao ldquoValue dis-tribution of meromorphic solutions of certain differencePainleve III equationsrdquo Advances in Difference Equationsvol 2018 p 171 2018

[14] A Ramani B Grammaticos T Tamizhmani andK M Tamizhmani ldquoe road to the discrete analogue of thePainleve property Nevanlinna meets singularity confine-mentrdquo Computers amp Mathematics with Applications vol 45no 6ndash9 pp 1001ndash1012 2003

[15] D C Barnett R G Halburd R J Korhonen andWMorganldquoNevanlinna theory for the $q$-difference operator andmeromorphic solutions of $q$-difference equationsrdquo Pro-ceedings of the Royal Society of Edinburgh Section A Math-ematics vol 137 no 3 pp 457ndash474 2007

[16] G G Gundersen J Heittokangas I Laine J Rieppo andD Yang ldquoMeromorphic solutions of generalized Schroderequationsrdquo Aequationes Mathematicae vol 63 no 1-2pp 110ndash135 2002

[17] I Laine and C-C Yang ldquoClunie theorems for difference andq-difference polynomialsrdquo Journal of the London Mathe-matical Society vol 76 no 3 pp 556ndash566 2007

[18] K Liu ldquoEntire solutions of Fermat type q-difference differ-ential equationsrdquo Electronic Journal of Differential Equationsvol 2013 no 59 pp 1ndash10 2013

[19] M Ru ldquoe recent progress in Nevanlinna theoryrdquo Journal ofJiangxi Normal University Natural Sciences Edition vol 42pp 1ndash11 2018

[20] H Y Xu S Y Liu and Q P Li ldquoEntire solutions for severalsystems of nonlinear difference and partial differential-dif-ference equations of Fermat-typerdquo Journal of MathematicalAnalysis and Applications vol 483 no 2 Article ID 1236412020

[21] H Y Xu S Y Liu and Q P Li ldquoe existence and growth ofsolutions for several systems of complex nonlinear differenceequationsrdquo Mediterranean Journal of Mathematics vol 16no 1 p 8 2019

[22] H Y Xu S Y Liu and X M Zheng ldquoSome properties ofmeromorphic solutions for q-difference equationsrdquo ElectronicJournal of Differential Equations vol 2017 no 175 pp 1ndash122017

[23] Q Y Yuan J R Long and D Z Qin ldquoe growth of solutionsof two certain types of q-difference differential equationsrdquoJournal of Jiangxi Normal University Natural Sciences Editionvol 44 pp 6ndash11 2020

[24] J Zhang and R Korhonen ldquoOn the Nevanlinna characteristicof f (qz) and its applicationsrdquo Journal of MathematicalAnalysis and Applications vol 369 no 2 pp 537ndash544 2010

[25] X M Zheng and Z X Chen ldquoOn properties of q-differenceequationsrdquo Acta Mathematica Scientia vol 32 no 2pp 724ndash734 2012

[26] X-M Zheng and Z-X Chen ldquoSome properties of mero-morphic solutions of q-difference equationsrdquo Journal ofMathematical Analysis and Applications vol 361 no 2pp 472ndash480 2010

[27] X G Qi and L Z Yang ldquoProperties of meromorphic solutionsof q-difference equationsrdquo Electronic Journal of DifferentialEquations vol 2015 no 59 pp 1ndash9 2015

[28] Y Liu and Y Zhang ldquoSome results of ϖ-Painleve differenceequationrdquo Advances in Difference Equations vol 2018 no 1p 282 2018

[29] W K Hayman Meromorphic Functions Clarendon PressOxford UK 1964

[30] L Yang Value Distribution 2eory Springer Berlin Ger-many 1993

[31] H X Yi and C C Yang Uniqueness 2eory of MeromorphicFunctions Kluwer Academic Publishers Dordrecht Neth-erland 2003


where η(z) λ(z) and υ(z) satisfy some conditions In theseequations equation (2) is called as the difference Painleve Iequation equation (3) is called as the difference Painleve IIequation and the last four equations are called as the dif-ference Painleve III equations

In the last decade or so there were a lot of papers fo-cusing on the properties of solutions for difference PainleveIndashIV equations (see [7ndash11]) For example Chen and Shon[12] in 2010 considered the difference Painleve I equation (3)and obtained the following theorem

Theorem 1 (see [12] eorem 4) Let a b c be constantswhere a b are not both equal to zero 2en the followingholds

(i) If ane 0 then (3) has no rational solution(ii) If a 0 and bne 0 then (3) has a nonzero constant

solution w(z) A where A satisfies 2A2 minus cAminus

b 0

e other rational solution w(z) satisfies w(z) (P(z)Q(z)) + A where P(z) and Q(z) are relatively primepolynomials and satisfy degPlt degQ

In 2013 and 2018 Zhang and Yi [11] and Du et al [13]studied the difference Painleve III equations with the con-stant coefficients and obtained the result as follows

Theorem 2 (see [11 13]) Iff is a transcendental finite-ordermeromorphic solution of

f(z + 1)f(z minus 1)(f(z) minus 1)2

f(z)2

minus λf(z) + μ (8)

where λ and μ are constants then the following holds

(i) τ(f) σ(f)(ii) If λμne 0 then λ(f) σ(f)(iii) For any η isin C 0 τ(f(z + η)) σ(f)(iv) λ(1Δf) λ(1(Δff)) σ(f)

Ramani et al [14] in 2003 investigated the existence oftranscendental solution of equation

(f(z + 1) + f(z))(f(z) + f(z minus 1)) R(z f)

P(z f)

Q(z f)

(9)

which is called as difference Painleve IV equations andobtained the result as follows

Theorem 3 (see [14]) If the second-order difference equation(9) admits a nonrational meromorphic solution of finiteorder then degzPle 4 and degzQle 2

Of late many mathematicians paid considerable atten-tion to the value distribution of solutions for complexq-difference equations which are formed by replacing theq-difference f(qz) q isin C 0 1 with f(z + c) of mero-morphic function in some expressions concerning complexdifference equations by utilizing the logarithmic derivative

lemma on q-difference operators given by Barnett et al [15]in 2007 (see [16ndash26]) For example Qi and Yang [27]considered the following equation

f(qz) + fz

q1113888 1113889

az + b

f(z)+ c (10)

which can be seen as q-difference analogues of (2) andobtained the result as follows

Theorem 4 (see [27] eorem 1) Let f(z) be a tran-scendental meromorphic solution with zero order of equation(10) and a b c be three constants such that a b cannot vanishsimultaneously 2en the following holds

(i) f(z) has infinitely many poles(ii) If ane 0 and any d isin C then f(z) minus d has infinitely

many zeros(iii) If a 0 and f(z) takes a finite value A finitely often

then A is a solution of 2z2 minus cz minus b 0

In 2018 Liu and Zhang [28] further investigated thefollowing equation

Y(ωz) + Y(z) + Yz

ω1113874 1113875

ξz + o

Y(z)+ ] (11)

and obtained the result as follows

Theorem 5 (see [28] eorem 1) Let Y(z) be a tran-scendental meromorphic solution with zero order of (11) andξ o ] be three constants such that ξ o cannot vanish si-multaneously 2en the following holds

(i) Y(z) has infinitely many poles(ii) For any finite value B if ξ 0 then Y(z) minus B has

infinitely many zeros(iii) If ξ 0 and Y(z) minus A has finite zeros then A is a

solution of 3z2 minus o minus ]z 0

Motivated by the idea [27 28] a natural question is whatis the result if we give q-difference analogues of (9) For thisquestion our main aim of this article is further to investigatesome properties of meromorphic solutions for someq-Painleve difference IV equations It seems that this topichas never been treated before

In what follows it should be assumed that the readers arefamiliar with the fundamental results and the standardnotations in the theory of Nevanlinna value distribution (seeHayman [29] Yang [30] and Yi and Yang [31]) Let f be ameromorphic function and we denote σ(f) λ(f) andλ(1f) to be the order the exponent of convergence of zerosand the exponent of convergence of poles of f(z) re-spectively and denote τ(f) to be the exponent of conver-gence of fixed points of f(z) which is defined by

τ(f) lim supr⟶+infin

logN(r 1(f(z) minus z))

log r (12)



lim supr⟶infin

1log r

1113946[1r]capF

1tdt (13)



[f(qz) + f(z)] f(z) + fz

q1113888 11138891113890 1113891 R(z)

A(z)

B(z)

(14)




2 (15)



2 (16)



[f(qz) + f(z)] f(z) + fz

q1113888 11138891113888 1113889 af(z)

2 (17)





[f(qz) + f(z)] f(z) + fz

q1113888 11138891113890 1113891 d(z)f(z) (18)








f(z) μ(z)

](z) 1113944

k

j1

cjλj

z minus zj1113872 1113873λj


z minus zj1113872 1113873

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦


(19)




μ(qz)

](qz)+μ(z)

](z)1113888 1113889

μ(z)

](z)+μ(zq)

](zq)1113888 1113889

A(z)

B(z) (20)


μ(qz)

](qz)⟶ 0

μ(z)

](z)⟶ 0

μ(zq)

](zq)⟶ 0

(21)


μ(qz)

](qz)⟶ α

μ(z)

](z)⟶ α

μ(zq)

](zq)⟶ α

(22)



f(z) aszs(1 + o(1))

f(qz) asqsz

s(1 + o(1))

fz

q1113888 1113889 asq

minus sz

s(1 + o(1))

A(z)

B(z) βz

aminus b(1 + o(1))

(23)


2 + qs

+ qminus s

( 1113857a2s z

2s(1 + o(1)) βz

aminus b(1 + o(1))

(24)


μ minus ] s

a minus b

2

(25)


B(z) β(1 + o(1)) (26)


qs

+ 2 + qminus s

1113858 1113859a2s z

2s β(1 + o(1)) (27)


μ minus ] 0

a minus b

2

(28)


μ(qz)

](qz)⟶ infin

μ(z)

](z)⟶ infin

μ(zq)

](zq)⟶ infin

(29)


μ(qz)

](qz)⟶ α

μ(z)

](z)⟶ α

μ(zq)

](zq)⟶ α

(30)


B(z) μ(qz)μ(z)]z

q1113888 1113889](z) + μ(qz)μ

z

q1113888 1113889](z)

21113890

+ μ(z)2](qz)]

z

q1113888 1113889 + μ(z)μ

z

q1113888 1113889](qz)](z)1113891

A(z)](z)2](qz)]

z

q1113888 1113889

(31)

Denote

A(z) ξaza


B(z) δbzb


μ(z) cμzμ


](z) ζ]z]


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(32)


qμminus ]

+ 2 + q]minus μ

( 1113857δbc2μζ

2]z


4]z

4]+a

(33)



2 (34)








T(r f) m(r f) + N(r f) (35)


m(r f) m(rinfin f)

12π

11139462π

0log+

f reiθ1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868dθ

(36)


N(r f) N(rinfin f)

1113946r

0

n(t f) minus n(0 f)

tdt + n(0 f)log r

(37)




m(r (1f minus a))

T(r f)


N(r (1f minus a))

T(r f)

(38)



log+T(r f)

log r


log+N(r 1f)

log r

λ1f

1113888 1113889 lim supr⟶+infin

log+N(r f)

log r

(39)


T r1

f minus a1113888 1113889 T(r f) + O(1)

m r 1113945

p

j1fj

⎛⎝ ⎞⎠le 1113944

p

j1m r fj1113872 1113873 m r 1113944

p

j1fj

⎛⎝ ⎞⎠

le 1113944

p

j1N r fj1113872 1113873 + logp

N r 1113945

p

j1fj

⎛⎝ ⎞⎠le 1113944

p

j1N r fj1113872 1113873 N r 1113944

p

j1fj

⎛⎝ ⎞⎠le 1113944

p

j1N r fj1113872 1113873

T r 1113945

p

j1fj

⎛⎝ ⎞⎠le 1113944

p

j1T r fj1113872 1113873 T r 1113944

p

j1fj

⎛⎝ ⎞⎠

le 1113944

p

j1T r fj1113872 1113873 + logp

(40)



m r1

f minus a1113888 1113889 S(r f) (41)




T(r f(qz)) (1 + o(1))T(r f(z))

N(r f(qz)) (1 + o(1))N(r f(z))(42)




P1(z f) ≔ [f(qz) + f(z)] f(z) + fz

q1113888 11138891113890 1113891 minus af(z)

2 equiv 0

(43)


P1(z d) 4d2

minus ad2 ne 0 (44)


N r1

f minus d1113888 1113889 T(r f) + S(r f) (45)




2 (46)



z

q1113888 11138891113890 1113891


(47)



z

q1113888 1113889 a[f(qz) minus f(z)]

(48)

that is


q1113888 1113889 0 (49)

Denote


+ Δqfz

q1113888 1113889 equiv 0

(50)


N r1Δqf minus d

1113888 1113889 T rΔqf1113872 1113873 + S rΔqf1113872 1113873 (51)




q1113888 11138891113890 1113891 af(ηz)

2

(52)


g1(qz) + g1(z)1113858 1113859 g1(z) + g1z

q1113888 11138891113890 1113891 ag1(z)

2 (53)

Set

P3 z g1( 1113857 ≔ g1(qz) + g1(z)1113858 1113859

middot g1(z) + g1z

q1113888 11138891113890 1113891 minus ag1(z)

2 equiv 0

(54)


N r1

f(ηz) minus z1113888 1113889 N r

1g1(z) minus z

1113888 1113889

T r g1(z)( 1113857 + S r g1( 1113857


T(r f) + S(r f)

(55)




q1113888 1113889 equiv 0 (56)


N r1

Δqf(ηz) minus z1113888 1113889 N r

1g2(z) minus z

1113888 1113889

T r g2(z)( 1113857 + S r g1( 1113857

T rΔqf(ηz)1113872 1113873 + S(r f(ηz))

(57)






m rf(qz)

f(z)1113888 1113889 S(r f) (58)




m r Pq(z f)1113872 1113873 S(r f) (60)


f(qz)

f(z)+ 11113888 1113889

f(zq)

f(z)+ 11113888 1113889

d(z)

f(z) (61)


T(r f(z)) T rd(z)

f(z)1113888 1113889 + O(log r)leT r

f(qz)

f(z)1113888 1113889

+ T rf(zq)

f(z)1113888 1113889 + O(log r)

T r g3(z)( 1113857 + T r g3z

q1113888 11138891113888 1113889 + O(log r)

2T r g3( 1113857 + S r g3( 1113857

2T rΔqf

f1113888 1113889 + S(r f)

(62)





f q2z1113872 1113873 + f(qz)

f(z) + f(zq)

d(qz)f(qz)

d(z)f(z) (64)


f(qz) g3(z)f(z)

f q2z1113872 1113873 g3(qz)f(qz)

g3(z)g3(qz)f(z)

fz

q1113888 1113889

f(z)

g3(zq)

(65)



f(z) + f(z)g3(zq)( 1113857

d(qz)

d(z)g3(z) (66)

that is

g3z

q1113888 1113889 g3(qz) + 1( 1113857

d(qz)

d(z)g

z

q1113888 1113889 + 11113888 1113889 (67)


N r g3z

q1113888 11138891113888 1113889 T r g3( 1113857 + S r g3( 1113857 (68)


(1 + o(1))N r g3( 1113857 N r g3z

q1113888 11138891113888 1113889

T r g3z

q1113888 11138891113888 1113889 + S r g3( 1113857

(1 + o(1))T r g3( 1113857 + S r g3( 1113857

(69)

is shows

N rΔqf

f1113888 1113889 N r g3( 1113857

T r g3( 1113857 + S r g3( 1113857

T rΔqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(70)


P5 z g3( 1113857 ≔ g3z

q1113888 1113889 g3(qz) + 1( 1113857

minusd(qz)

d(z)g

z

q1113888 1113889 + 11113888 1113889 equiv 0

(71)



P5(z ϱ + 1) (ϱ + 1)(ϱ + 1 + 1) minusd(qz)

d(z)(ϱ + 1 + 1)


d(z)1113888 1113889

(72)


m r1

Δqff1113872 1113873 minus ϱ⎛⎝ ⎞⎠ m r


1113888 1113889

S r g3( 1113857 S rΔqf

f1113888 1113889

(73)

is means

N r1

Δqff1113872 1113873 minus ϱ⎛⎝ ⎞⎠ T r

Δqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(74)



m r1

f(qz) + f(z)1113888 1113889 m r

f(z) + f(zq)

d(z)f(z)1113888 1113889

lem r1

d(z)1113888 1113889 + m r

f(z) + f(zq)

f(z)1113888 1113889

+ O(1)

le S(r f)

(75)


N r1

f(qz) + f(z)1113888 1113889 T r

1f(qz) + f(z)

1113888 1113889 + S(r f)

T r1

d(z)

f(z) + f(zq)

f(z)1113888 1113889 + S(r f)

T rf(zq)

f(z)1113888 1113889 + O(log r) + S(r f)

T rf(qz)

f(z)1113888 1113889 + S(r f)

T rΔqf

f1113888 1113889 + S(r f)

(76)




N r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ N r

1f(qz) + f(z)

1113888 1113889

T rΔqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(77)




m r1f

1113888 1113889lem r1

d(z)1113888 1113889 + m r

f(qz)

f(z)1113888 1113889 + m r

f(zq)

f(z)1113888 1113889

S(r f)

(78)

which implies

N r1f

1113888 1113889 T(r f) + S(r f) (79)


P6(z f) ≔ [f(qz) + f(z)] f(z) + fz


(80)



ϱ(4ϱ minus d(z))≢ 0(81)



N r1

f minus ϱ1113888 1113889 T(r f) + S(r f) (82)


m r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ S r

Δqf

f1113888 1113889

S(r f)

(83)



m r f(z) + fz

q1113888 11138891113888 1113889 m r

d(z)f(z)

f(qz) + f(z)1113888 1113889

m r1

(f(qz)f(z)) + 11113888 1113889 + O(log r)

m r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ + O(log r)

S(r f)

(84)



q1113888 11138891113888 1113889 + m(r f(qz) + f(z))

+ O(log r)

le 2m r f(z) + fz

q1113888 11138891113888 1113889 + m r

f(qz) + f(z)

f(z) + f(zq)1113888 1113889

+ O(log r)

S(r f)

(85)

which implies

N(r f) T(r f) + S(r f) (86)




q1113888 11138891113890 1113891 d(z)f(z)

(87)


N r1f

1113888 1113889leN r1Δqf(z)

1113888 1113889 + N r1

Δqf(zq)1113888 1113889 + N(r d)

2N r1Δqf

1113888 1113889 + S(r f)

(88)




f1113888 1113889 S(r f)

S rΔqf1113872 1113873

(89)

which implies


T r1Δqf

1113888 1113889 + S rΔqf1113872 1113873

(90)



P5(z z + 1) z

q+ 11113888 1113889(qz + 1 + 1) minus

d(qz)

d(z)

z

q+ 1 + 11113888 1113889

(z + q)(qz + 2)

qminus

d(qz)

d(z)

z + 2q

q

(91)


m r1

Δqff1113872 1113873 minus z⎛⎝ ⎞⎠ m r


1113888 1113889

S r g3( 1113857

S rΔqf

f1113888 1113889

(92)

is leads to

N r1

Δqff1113872 1113873 minus z⎛⎝ ⎞⎠ T r

Δqf

f1113888 1113889 + S r

Δqf

f1113888 1113889 (93)



Data Availability







Acknowledgments


References


































lim supr⟶infin

1log r

1113946[1r]capF

1tdt (13)



[f(qz) + f(z)] f(z) + fz

q1113888 11138891113890 1113891 R(z)

A(z)

B(z)

(14)




2 (15)



2 (16)



[f(qz) + f(z)] f(z) + fz

q1113888 11138891113888 1113889 af(z)

2 (17)





[f(qz) + f(z)] f(z) + fz

q1113888 11138891113890 1113891 d(z)f(z) (18)








f(z) μ(z)

](z) 1113944

k

j1

cjλj

z minus zj1113872 1113873λj


z minus zj1113872 1113873

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦


(19)




μ(qz)

](qz)+μ(z)

](z)1113888 1113889

μ(z)

](z)+μ(zq)

](zq)1113888 1113889

A(z)

B(z) (20)


μ(qz)

](qz)⟶ 0

μ(z)

](z)⟶ 0

μ(zq)

](zq)⟶ 0

(21)


μ(qz)

](qz)⟶ α

μ(z)

](z)⟶ α

μ(zq)

](zq)⟶ α

(22)



f(z) aszs(1 + o(1))

f(qz) asqsz

s(1 + o(1))

fz

q1113888 1113889 asq

minus sz

s(1 + o(1))

A(z)

B(z) βz

aminus b(1 + o(1))

(23)


2 + qs

+ qminus s

( 1113857a2s z

2s(1 + o(1)) βz

aminus b(1 + o(1))

(24)


μ minus ] s

a minus b

2

(25)


B(z) β(1 + o(1)) (26)


qs

+ 2 + qminus s

1113858 1113859a2s z

2s β(1 + o(1)) (27)


μ minus ] 0

a minus b

2

(28)


μ(qz)

](qz)⟶ infin

μ(z)

](z)⟶ infin

μ(zq)

](zq)⟶ infin

(29)


μ(qz)

](qz)⟶ α

μ(z)

](z)⟶ α

μ(zq)

](zq)⟶ α

(30)


B(z) μ(qz)μ(z)]z

q1113888 1113889](z) + μ(qz)μ

z

q1113888 1113889](z)

21113890

+ μ(z)2](qz)]

z

q1113888 1113889 + μ(z)μ

z

q1113888 1113889](qz)](z)1113891

A(z)](z)2](qz)]

z

q1113888 1113889

(31)

Denote

A(z) ξaza


B(z) δbzb


μ(z) cμzμ


](z) ζ]z]


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(32)


qμminus ]

+ 2 + q]minus μ

( 1113857δbc2μζ

2]z


4]z

4]+a

(33)



2 (34)








T(r f) m(r f) + N(r f) (35)


m(r f) m(rinfin f)

12π

11139462π

0log+

f reiθ1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868dθ

(36)


N(r f) N(rinfin f)

1113946r

0

n(t f) minus n(0 f)

tdt + n(0 f)log r

(37)




m(r (1f minus a))

T(r f)


N(r (1f minus a))

T(r f)

(38)



log+T(r f)

log r


log+N(r 1f)

log r

λ1f

1113888 1113889 lim supr⟶+infin

log+N(r f)

log r

(39)


T r1

f minus a1113888 1113889 T(r f) + O(1)

m r 1113945

p

j1fj

⎛⎝ ⎞⎠le 1113944

p

j1m r fj1113872 1113873 m r 1113944

p

j1fj

⎛⎝ ⎞⎠

le 1113944

p

j1N r fj1113872 1113873 + logp

N r 1113945

p

j1fj

⎛⎝ ⎞⎠le 1113944

p

j1N r fj1113872 1113873 N r 1113944

p

j1fj

⎛⎝ ⎞⎠le 1113944

p

j1N r fj1113872 1113873

T r 1113945

p

j1fj

⎛⎝ ⎞⎠le 1113944

p

j1T r fj1113872 1113873 T r 1113944

p

j1fj

⎛⎝ ⎞⎠

le 1113944

p

j1T r fj1113872 1113873 + logp

(40)



m r1

f minus a1113888 1113889 S(r f) (41)




T(r f(qz)) (1 + o(1))T(r f(z))

N(r f(qz)) (1 + o(1))N(r f(z))(42)




P1(z f) ≔ [f(qz) + f(z)] f(z) + fz

q1113888 11138891113890 1113891 minus af(z)

2 equiv 0

(43)


P1(z d) 4d2

minus ad2 ne 0 (44)


N r1

f minus d1113888 1113889 T(r f) + S(r f) (45)




2 (46)



z

q1113888 11138891113890 1113891


(47)



z

q1113888 1113889 a[f(qz) minus f(z)]

(48)

that is


q1113888 1113889 0 (49)

Denote


+ Δqfz

q1113888 1113889 equiv 0

(50)


N r1Δqf minus d

1113888 1113889 T rΔqf1113872 1113873 + S rΔqf1113872 1113873 (51)




q1113888 11138891113890 1113891 af(ηz)

2

(52)


g1(qz) + g1(z)1113858 1113859 g1(z) + g1z

q1113888 11138891113890 1113891 ag1(z)

2 (53)

Set

P3 z g1( 1113857 ≔ g1(qz) + g1(z)1113858 1113859

middot g1(z) + g1z

q1113888 11138891113890 1113891 minus ag1(z)

2 equiv 0

(54)


N r1

f(ηz) minus z1113888 1113889 N r

1g1(z) minus z

1113888 1113889

T r g1(z)( 1113857 + S r g1( 1113857


T(r f) + S(r f)

(55)




q1113888 1113889 equiv 0 (56)


N r1

Δqf(ηz) minus z1113888 1113889 N r

1g2(z) minus z

1113888 1113889

T r g2(z)( 1113857 + S r g1( 1113857

T rΔqf(ηz)1113872 1113873 + S(r f(ηz))

(57)






m rf(qz)

f(z)1113888 1113889 S(r f) (58)




m r Pq(z f)1113872 1113873 S(r f) (60)


f(qz)

f(z)+ 11113888 1113889

f(zq)

f(z)+ 11113888 1113889

d(z)

f(z) (61)


T(r f(z)) T rd(z)

f(z)1113888 1113889 + O(log r)leT r

f(qz)

f(z)1113888 1113889

+ T rf(zq)

f(z)1113888 1113889 + O(log r)

T r g3(z)( 1113857 + T r g3z

q1113888 11138891113888 1113889 + O(log r)

2T r g3( 1113857 + S r g3( 1113857

2T rΔqf

f1113888 1113889 + S(r f)

(62)





f q2z1113872 1113873 + f(qz)

f(z) + f(zq)

d(qz)f(qz)

d(z)f(z) (64)


f(qz) g3(z)f(z)

f q2z1113872 1113873 g3(qz)f(qz)

g3(z)g3(qz)f(z)

fz

q1113888 1113889

f(z)

g3(zq)

(65)



f(z) + f(z)g3(zq)( 1113857

d(qz)

d(z)g3(z) (66)

that is

g3z

q1113888 1113889 g3(qz) + 1( 1113857

d(qz)

d(z)g

z

q1113888 1113889 + 11113888 1113889 (67)


N r g3z

q1113888 11138891113888 1113889 T r g3( 1113857 + S r g3( 1113857 (68)


(1 + o(1))N r g3( 1113857 N r g3z

q1113888 11138891113888 1113889

T r g3z

q1113888 11138891113888 1113889 + S r g3( 1113857

(1 + o(1))T r g3( 1113857 + S r g3( 1113857

(69)

is shows

N rΔqf

f1113888 1113889 N r g3( 1113857

T r g3( 1113857 + S r g3( 1113857

T rΔqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(70)


P5 z g3( 1113857 ≔ g3z

q1113888 1113889 g3(qz) + 1( 1113857

minusd(qz)

d(z)g

z

q1113888 1113889 + 11113888 1113889 equiv 0

(71)



P5(z ϱ + 1) (ϱ + 1)(ϱ + 1 + 1) minusd(qz)

d(z)(ϱ + 1 + 1)


d(z)1113888 1113889

(72)


m r1

Δqff1113872 1113873 minus ϱ⎛⎝ ⎞⎠ m r


1113888 1113889

S r g3( 1113857 S rΔqf

f1113888 1113889

(73)

is means

N r1

Δqff1113872 1113873 minus ϱ⎛⎝ ⎞⎠ T r

Δqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(74)



m r1

f(qz) + f(z)1113888 1113889 m r

f(z) + f(zq)

d(z)f(z)1113888 1113889

lem r1

d(z)1113888 1113889 + m r

f(z) + f(zq)

f(z)1113888 1113889

+ O(1)

le S(r f)

(75)


N r1

f(qz) + f(z)1113888 1113889 T r

1f(qz) + f(z)

1113888 1113889 + S(r f)

T r1

d(z)

f(z) + f(zq)

f(z)1113888 1113889 + S(r f)

T rf(zq)

f(z)1113888 1113889 + O(log r) + S(r f)

T rf(qz)

f(z)1113888 1113889 + S(r f)

T rΔqf

f1113888 1113889 + S(r f)

(76)




N r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ N r

1f(qz) + f(z)

1113888 1113889

T rΔqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(77)




m r1f

1113888 1113889lem r1

d(z)1113888 1113889 + m r

f(qz)

f(z)1113888 1113889 + m r

f(zq)

f(z)1113888 1113889

S(r f)

(78)

which implies

N r1f

1113888 1113889 T(r f) + S(r f) (79)


P6(z f) ≔ [f(qz) + f(z)] f(z) + fz


(80)



ϱ(4ϱ minus d(z))≢ 0(81)



N r1

f minus ϱ1113888 1113889 T(r f) + S(r f) (82)


m r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ S r

Δqf

f1113888 1113889

S(r f)

(83)



m r f(z) + fz

q1113888 11138891113888 1113889 m r

d(z)f(z)

f(qz) + f(z)1113888 1113889

m r1

(f(qz)f(z)) + 11113888 1113889 + O(log r)

m r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ + O(log r)

S(r f)

(84)



q1113888 11138891113888 1113889 + m(r f(qz) + f(z))

+ O(log r)

le 2m r f(z) + fz

q1113888 11138891113888 1113889 + m r

f(qz) + f(z)

f(z) + f(zq)1113888 1113889

+ O(log r)

S(r f)

(85)

which implies

N(r f) T(r f) + S(r f) (86)




q1113888 11138891113890 1113891 d(z)f(z)

(87)


N r1f

1113888 1113889leN r1Δqf(z)

1113888 1113889 + N r1

Δqf(zq)1113888 1113889 + N(r d)

2N r1Δqf

1113888 1113889 + S(r f)

(88)




f1113888 1113889 S(r f)

S rΔqf1113872 1113873

(89)

which implies


T r1Δqf

1113888 1113889 + S rΔqf1113872 1113873

(90)



P5(z z + 1) z

q+ 11113888 1113889(qz + 1 + 1) minus

d(qz)

d(z)

z

q+ 1 + 11113888 1113889

(z + q)(qz + 2)

qminus

d(qz)

d(z)

z + 2q

q

(91)


m r1

Δqff1113872 1113873 minus z⎛⎝ ⎞⎠ m r


1113888 1113889

S r g3( 1113857

S rΔqf

f1113888 1113889

(92)

is leads to

N r1

Δqff1113872 1113873 minus z⎛⎝ ⎞⎠ T r

Δqf

f1113888 1113889 + S r

Δqf

f1113888 1113889 (93)



Data Availability







Acknowledgments


References


































f(z) aszs(1 + o(1))

f(qz) asqsz

s(1 + o(1))

fz

q1113888 1113889 asq

minus sz

s(1 + o(1))

A(z)

B(z) βz

aminus b(1 + o(1))

(23)


2 + qs

+ qminus s

( 1113857a2s z

2s(1 + o(1)) βz

aminus b(1 + o(1))

(24)


μ minus ] s

a minus b

2

(25)


B(z) β(1 + o(1)) (26)


qs

+ 2 + qminus s

1113858 1113859a2s z

2s β(1 + o(1)) (27)


μ minus ] 0

a minus b

2

(28)


μ(qz)

](qz)⟶ infin

μ(z)

](z)⟶ infin

μ(zq)

](zq)⟶ infin

(29)


μ(qz)

](qz)⟶ α

μ(z)

](z)⟶ α

μ(zq)

](zq)⟶ α

(30)


B(z) μ(qz)μ(z)]z

q1113888 1113889](z) + μ(qz)μ

z

q1113888 1113889](z)

21113890

+ μ(z)2](qz)]

z

q1113888 1113889 + μ(z)μ

z

q1113888 1113889](qz)](z)1113891

A(z)](z)2](qz)]

z

q1113888 1113889

(31)

Denote

A(z) ξaza


B(z) δbzb


μ(z) cμzμ


](z) ζ]z]


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(32)


qμminus ]

+ 2 + q]minus μ

( 1113857δbc2μζ

2]z


4]z

4]+a

(33)



2 (34)








T(r f) m(r f) + N(r f) (35)


m(r f) m(rinfin f)

12π

11139462π

0log+

f reiθ1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868dθ

(36)


N(r f) N(rinfin f)

1113946r

0

n(t f) minus n(0 f)

tdt + n(0 f)log r

(37)




m(r (1f minus a))

T(r f)


N(r (1f minus a))

T(r f)

(38)



log+T(r f)

log r


log+N(r 1f)

log r

λ1f

1113888 1113889 lim supr⟶+infin

log+N(r f)

log r

(39)


T r1

f minus a1113888 1113889 T(r f) + O(1)

m r 1113945

p

j1fj

⎛⎝ ⎞⎠le 1113944

p

j1m r fj1113872 1113873 m r 1113944

p

j1fj

⎛⎝ ⎞⎠

le 1113944

p

j1N r fj1113872 1113873 + logp

N r 1113945

p

j1fj

⎛⎝ ⎞⎠le 1113944

p

j1N r fj1113872 1113873 N r 1113944

p

j1fj

⎛⎝ ⎞⎠le 1113944

p

j1N r fj1113872 1113873

T r 1113945

p

j1fj

⎛⎝ ⎞⎠le 1113944

p

j1T r fj1113872 1113873 T r 1113944

p

j1fj

⎛⎝ ⎞⎠

le 1113944

p

j1T r fj1113872 1113873 + logp

(40)



m r1

f minus a1113888 1113889 S(r f) (41)




T(r f(qz)) (1 + o(1))T(r f(z))

N(r f(qz)) (1 + o(1))N(r f(z))(42)




P1(z f) ≔ [f(qz) + f(z)] f(z) + fz

q1113888 11138891113890 1113891 minus af(z)

2 equiv 0

(43)


P1(z d) 4d2

minus ad2 ne 0 (44)


N r1

f minus d1113888 1113889 T(r f) + S(r f) (45)




2 (46)



z

q1113888 11138891113890 1113891


(47)



z

q1113888 1113889 a[f(qz) minus f(z)]

(48)

that is


q1113888 1113889 0 (49)

Denote


+ Δqfz

q1113888 1113889 equiv 0

(50)


N r1Δqf minus d

1113888 1113889 T rΔqf1113872 1113873 + S rΔqf1113872 1113873 (51)




q1113888 11138891113890 1113891 af(ηz)

2

(52)


g1(qz) + g1(z)1113858 1113859 g1(z) + g1z

q1113888 11138891113890 1113891 ag1(z)

2 (53)

Set

P3 z g1( 1113857 ≔ g1(qz) + g1(z)1113858 1113859

middot g1(z) + g1z

q1113888 11138891113890 1113891 minus ag1(z)

2 equiv 0

(54)


N r1

f(ηz) minus z1113888 1113889 N r

1g1(z) minus z

1113888 1113889

T r g1(z)( 1113857 + S r g1( 1113857


T(r f) + S(r f)

(55)




q1113888 1113889 equiv 0 (56)


N r1

Δqf(ηz) minus z1113888 1113889 N r

1g2(z) minus z

1113888 1113889

T r g2(z)( 1113857 + S r g1( 1113857

T rΔqf(ηz)1113872 1113873 + S(r f(ηz))

(57)






m rf(qz)

f(z)1113888 1113889 S(r f) (58)




m r Pq(z f)1113872 1113873 S(r f) (60)


f(qz)

f(z)+ 11113888 1113889

f(zq)

f(z)+ 11113888 1113889

d(z)

f(z) (61)


T(r f(z)) T rd(z)

f(z)1113888 1113889 + O(log r)leT r

f(qz)

f(z)1113888 1113889

+ T rf(zq)

f(z)1113888 1113889 + O(log r)

T r g3(z)( 1113857 + T r g3z

q1113888 11138891113888 1113889 + O(log r)

2T r g3( 1113857 + S r g3( 1113857

2T rΔqf

f1113888 1113889 + S(r f)

(62)





f q2z1113872 1113873 + f(qz)

f(z) + f(zq)

d(qz)f(qz)

d(z)f(z) (64)


f(qz) g3(z)f(z)

f q2z1113872 1113873 g3(qz)f(qz)

g3(z)g3(qz)f(z)

fz

q1113888 1113889

f(z)

g3(zq)

(65)



f(z) + f(z)g3(zq)( 1113857

d(qz)

d(z)g3(z) (66)

that is

g3z

q1113888 1113889 g3(qz) + 1( 1113857

d(qz)

d(z)g

z

q1113888 1113889 + 11113888 1113889 (67)


N r g3z

q1113888 11138891113888 1113889 T r g3( 1113857 + S r g3( 1113857 (68)


(1 + o(1))N r g3( 1113857 N r g3z

q1113888 11138891113888 1113889

T r g3z

q1113888 11138891113888 1113889 + S r g3( 1113857

(1 + o(1))T r g3( 1113857 + S r g3( 1113857

(69)

is shows

N rΔqf

f1113888 1113889 N r g3( 1113857

T r g3( 1113857 + S r g3( 1113857

T rΔqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(70)


P5 z g3( 1113857 ≔ g3z

q1113888 1113889 g3(qz) + 1( 1113857

minusd(qz)

d(z)g

z

q1113888 1113889 + 11113888 1113889 equiv 0

(71)



P5(z ϱ + 1) (ϱ + 1)(ϱ + 1 + 1) minusd(qz)

d(z)(ϱ + 1 + 1)


d(z)1113888 1113889

(72)


m r1

Δqff1113872 1113873 minus ϱ⎛⎝ ⎞⎠ m r


1113888 1113889

S r g3( 1113857 S rΔqf

f1113888 1113889

(73)

is means

N r1

Δqff1113872 1113873 minus ϱ⎛⎝ ⎞⎠ T r

Δqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(74)



m r1

f(qz) + f(z)1113888 1113889 m r

f(z) + f(zq)

d(z)f(z)1113888 1113889

lem r1

d(z)1113888 1113889 + m r

f(z) + f(zq)

f(z)1113888 1113889

+ O(1)

le S(r f)

(75)


N r1

f(qz) + f(z)1113888 1113889 T r

1f(qz) + f(z)

1113888 1113889 + S(r f)

T r1

d(z)

f(z) + f(zq)

f(z)1113888 1113889 + S(r f)

T rf(zq)

f(z)1113888 1113889 + O(log r) + S(r f)

T rf(qz)

f(z)1113888 1113889 + S(r f)

T rΔqf

f1113888 1113889 + S(r f)

(76)




N r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ N r

1f(qz) + f(z)

1113888 1113889

T rΔqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(77)




m r1f

1113888 1113889lem r1

d(z)1113888 1113889 + m r

f(qz)

f(z)1113888 1113889 + m r

f(zq)

f(z)1113888 1113889

S(r f)

(78)

which implies

N r1f

1113888 1113889 T(r f) + S(r f) (79)


P6(z f) ≔ [f(qz) + f(z)] f(z) + fz


(80)



ϱ(4ϱ minus d(z))≢ 0(81)



N r1

f minus ϱ1113888 1113889 T(r f) + S(r f) (82)


m r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ S r

Δqf

f1113888 1113889

S(r f)

(83)



m r f(z) + fz

q1113888 11138891113888 1113889 m r

d(z)f(z)

f(qz) + f(z)1113888 1113889

m r1

(f(qz)f(z)) + 11113888 1113889 + O(log r)

m r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ + O(log r)

S(r f)

(84)



q1113888 11138891113888 1113889 + m(r f(qz) + f(z))

+ O(log r)

le 2m r f(z) + fz

q1113888 11138891113888 1113889 + m r

f(qz) + f(z)

f(z) + f(zq)1113888 1113889

+ O(log r)

S(r f)

(85)

which implies

N(r f) T(r f) + S(r f) (86)




q1113888 11138891113890 1113891 d(z)f(z)

(87)


N r1f

1113888 1113889leN r1Δqf(z)

1113888 1113889 + N r1

Δqf(zq)1113888 1113889 + N(r d)

2N r1Δqf

1113888 1113889 + S(r f)

(88)




f1113888 1113889 S(r f)

S rΔqf1113872 1113873

(89)

which implies


T r1Δqf

1113888 1113889 + S rΔqf1113872 1113873

(90)



P5(z z + 1) z

q+ 11113888 1113889(qz + 1 + 1) minus

d(qz)

d(z)

z

q+ 1 + 11113888 1113889

(z + q)(qz + 2)

qminus

d(qz)

d(z)

z + 2q

q

(91)


m r1

Δqff1113872 1113873 minus z⎛⎝ ⎞⎠ m r


1113888 1113889

S r g3( 1113857

S rΔqf

f1113888 1113889

(92)

is leads to

N r1

Δqff1113872 1113873 minus z⎛⎝ ⎞⎠ T r

Δqf

f1113888 1113889 + S r

Δqf

f1113888 1113889 (93)



Data Availability







Acknowledgments


References




































T(r f) m(r f) + N(r f) (35)


m(r f) m(rinfin f)

12π

11139462π

0log+

f reiθ1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868dθ

(36)


N(r f) N(rinfin f)

1113946r

0

n(t f) minus n(0 f)

tdt + n(0 f)log r

(37)




m(r (1f minus a))

T(r f)


N(r (1f minus a))

T(r f)

(38)



log+T(r f)

log r


log+N(r 1f)

log r

λ1f

1113888 1113889 lim supr⟶+infin

log+N(r f)

log r

(39)


T r1

f minus a1113888 1113889 T(r f) + O(1)

m r 1113945

p

j1fj

⎛⎝ ⎞⎠le 1113944

p

j1m r fj1113872 1113873 m r 1113944

p

j1fj

⎛⎝ ⎞⎠

le 1113944

p

j1N r fj1113872 1113873 + logp

N r 1113945

p

j1fj

⎛⎝ ⎞⎠le 1113944

p

j1N r fj1113872 1113873 N r 1113944

p

j1fj

⎛⎝ ⎞⎠le 1113944

p

j1N r fj1113872 1113873

T r 1113945

p

j1fj

⎛⎝ ⎞⎠le 1113944

p

j1T r fj1113872 1113873 T r 1113944

p

j1fj

⎛⎝ ⎞⎠

le 1113944

p

j1T r fj1113872 1113873 + logp

(40)



m r1

f minus a1113888 1113889 S(r f) (41)




T(r f(qz)) (1 + o(1))T(r f(z))

N(r f(qz)) (1 + o(1))N(r f(z))(42)




P1(z f) ≔ [f(qz) + f(z)] f(z) + fz

q1113888 11138891113890 1113891 minus af(z)

2 equiv 0

(43)


P1(z d) 4d2

minus ad2 ne 0 (44)


N r1

f minus d1113888 1113889 T(r f) + S(r f) (45)




2 (46)



z

q1113888 11138891113890 1113891


(47)



z

q1113888 1113889 a[f(qz) minus f(z)]

(48)

that is


q1113888 1113889 0 (49)

Denote


+ Δqfz

q1113888 1113889 equiv 0

(50)


N r1Δqf minus d

1113888 1113889 T rΔqf1113872 1113873 + S rΔqf1113872 1113873 (51)




q1113888 11138891113890 1113891 af(ηz)

2

(52)


g1(qz) + g1(z)1113858 1113859 g1(z) + g1z

q1113888 11138891113890 1113891 ag1(z)

2 (53)

Set

P3 z g1( 1113857 ≔ g1(qz) + g1(z)1113858 1113859

middot g1(z) + g1z

q1113888 11138891113890 1113891 minus ag1(z)

2 equiv 0

(54)


N r1

f(ηz) minus z1113888 1113889 N r

1g1(z) minus z

1113888 1113889

T r g1(z)( 1113857 + S r g1( 1113857


T(r f) + S(r f)

(55)




q1113888 1113889 equiv 0 (56)


N r1

Δqf(ηz) minus z1113888 1113889 N r

1g2(z) minus z

1113888 1113889

T r g2(z)( 1113857 + S r g1( 1113857

T rΔqf(ηz)1113872 1113873 + S(r f(ηz))

(57)






m rf(qz)

f(z)1113888 1113889 S(r f) (58)




m r Pq(z f)1113872 1113873 S(r f) (60)


f(qz)

f(z)+ 11113888 1113889

f(zq)

f(z)+ 11113888 1113889

d(z)

f(z) (61)


T(r f(z)) T rd(z)

f(z)1113888 1113889 + O(log r)leT r

f(qz)

f(z)1113888 1113889

+ T rf(zq)

f(z)1113888 1113889 + O(log r)

T r g3(z)( 1113857 + T r g3z

q1113888 11138891113888 1113889 + O(log r)

2T r g3( 1113857 + S r g3( 1113857

2T rΔqf

f1113888 1113889 + S(r f)

(62)





f q2z1113872 1113873 + f(qz)

f(z) + f(zq)

d(qz)f(qz)

d(z)f(z) (64)


f(qz) g3(z)f(z)

f q2z1113872 1113873 g3(qz)f(qz)

g3(z)g3(qz)f(z)

fz

q1113888 1113889

f(z)

g3(zq)

(65)



f(z) + f(z)g3(zq)( 1113857

d(qz)

d(z)g3(z) (66)

that is

g3z

q1113888 1113889 g3(qz) + 1( 1113857

d(qz)

d(z)g

z

q1113888 1113889 + 11113888 1113889 (67)


N r g3z

q1113888 11138891113888 1113889 T r g3( 1113857 + S r g3( 1113857 (68)


(1 + o(1))N r g3( 1113857 N r g3z

q1113888 11138891113888 1113889

T r g3z

q1113888 11138891113888 1113889 + S r g3( 1113857

(1 + o(1))T r g3( 1113857 + S r g3( 1113857

(69)

is shows

N rΔqf

f1113888 1113889 N r g3( 1113857

T r g3( 1113857 + S r g3( 1113857

T rΔqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(70)


P5 z g3( 1113857 ≔ g3z

q1113888 1113889 g3(qz) + 1( 1113857

minusd(qz)

d(z)g

z

q1113888 1113889 + 11113888 1113889 equiv 0

(71)



P5(z ϱ + 1) (ϱ + 1)(ϱ + 1 + 1) minusd(qz)

d(z)(ϱ + 1 + 1)


d(z)1113888 1113889

(72)


m r1

Δqff1113872 1113873 minus ϱ⎛⎝ ⎞⎠ m r


1113888 1113889

S r g3( 1113857 S rΔqf

f1113888 1113889

(73)

is means

N r1

Δqff1113872 1113873 minus ϱ⎛⎝ ⎞⎠ T r

Δqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(74)



m r1

f(qz) + f(z)1113888 1113889 m r

f(z) + f(zq)

d(z)f(z)1113888 1113889

lem r1

d(z)1113888 1113889 + m r

f(z) + f(zq)

f(z)1113888 1113889

+ O(1)

le S(r f)

(75)


N r1

f(qz) + f(z)1113888 1113889 T r

1f(qz) + f(z)

1113888 1113889 + S(r f)

T r1

d(z)

f(z) + f(zq)

f(z)1113888 1113889 + S(r f)

T rf(zq)

f(z)1113888 1113889 + O(log r) + S(r f)

T rf(qz)

f(z)1113888 1113889 + S(r f)

T rΔqf

f1113888 1113889 + S(r f)

(76)




N r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ N r

1f(qz) + f(z)

1113888 1113889

T rΔqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(77)




m r1f

1113888 1113889lem r1

d(z)1113888 1113889 + m r

f(qz)

f(z)1113888 1113889 + m r

f(zq)

f(z)1113888 1113889

S(r f)

(78)

which implies

N r1f

1113888 1113889 T(r f) + S(r f) (79)


P6(z f) ≔ [f(qz) + f(z)] f(z) + fz


(80)



ϱ(4ϱ minus d(z))≢ 0(81)



N r1

f minus ϱ1113888 1113889 T(r f) + S(r f) (82)


m r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ S r

Δqf

f1113888 1113889

S(r f)

(83)



m r f(z) + fz

q1113888 11138891113888 1113889 m r

d(z)f(z)

f(qz) + f(z)1113888 1113889

m r1

(f(qz)f(z)) + 11113888 1113889 + O(log r)

m r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ + O(log r)

S(r f)

(84)



q1113888 11138891113888 1113889 + m(r f(qz) + f(z))

+ O(log r)

le 2m r f(z) + fz

q1113888 11138891113888 1113889 + m r

f(qz) + f(z)

f(z) + f(zq)1113888 1113889

+ O(log r)

S(r f)

(85)

which implies

N(r f) T(r f) + S(r f) (86)




q1113888 11138891113890 1113891 d(z)f(z)

(87)


N r1f

1113888 1113889leN r1Δqf(z)

1113888 1113889 + N r1

Δqf(zq)1113888 1113889 + N(r d)

2N r1Δqf

1113888 1113889 + S(r f)

(88)




f1113888 1113889 S(r f)

S rΔqf1113872 1113873

(89)

which implies


T r1Δqf

1113888 1113889 + S rΔqf1113872 1113873

(90)



P5(z z + 1) z

q+ 11113888 1113889(qz + 1 + 1) minus

d(qz)

d(z)

z

q+ 1 + 11113888 1113889

(z + q)(qz + 2)

qminus

d(qz)

d(z)

z + 2q

q

(91)


m r1

Δqff1113872 1113873 minus z⎛⎝ ⎞⎠ m r


1113888 1113889

S r g3( 1113857

S rΔqf

f1113888 1113889

(92)

is leads to

N r1

Δqff1113872 1113873 minus z⎛⎝ ⎞⎠ T r

Δqf

f1113888 1113889 + S r

Δqf

f1113888 1113889 (93)



Data Availability







Acknowledgments


References


































P1(z f) ≔ [f(qz) + f(z)] f(z) + fz

q1113888 11138891113890 1113891 minus af(z)

2 equiv 0

(43)


P1(z d) 4d2

minus ad2 ne 0 (44)


N r1

f minus d1113888 1113889 T(r f) + S(r f) (45)




2 (46)



z

q1113888 11138891113890 1113891


(47)



z

q1113888 1113889 a[f(qz) minus f(z)]

(48)

that is


q1113888 1113889 0 (49)

Denote


+ Δqfz

q1113888 1113889 equiv 0

(50)


N r1Δqf minus d

1113888 1113889 T rΔqf1113872 1113873 + S rΔqf1113872 1113873 (51)




q1113888 11138891113890 1113891 af(ηz)

2

(52)


g1(qz) + g1(z)1113858 1113859 g1(z) + g1z

q1113888 11138891113890 1113891 ag1(z)

2 (53)

Set

P3 z g1( 1113857 ≔ g1(qz) + g1(z)1113858 1113859

middot g1(z) + g1z

q1113888 11138891113890 1113891 minus ag1(z)

2 equiv 0

(54)


N r1

f(ηz) minus z1113888 1113889 N r

1g1(z) minus z

1113888 1113889

T r g1(z)( 1113857 + S r g1( 1113857


T(r f) + S(r f)

(55)




q1113888 1113889 equiv 0 (56)


N r1

Δqf(ηz) minus z1113888 1113889 N r

1g2(z) minus z

1113888 1113889

T r g2(z)( 1113857 + S r g1( 1113857

T rΔqf(ηz)1113872 1113873 + S(r f(ηz))

(57)






m rf(qz)

f(z)1113888 1113889 S(r f) (58)




m r Pq(z f)1113872 1113873 S(r f) (60)


f(qz)

f(z)+ 11113888 1113889

f(zq)

f(z)+ 11113888 1113889

d(z)

f(z) (61)


T(r f(z)) T rd(z)

f(z)1113888 1113889 + O(log r)leT r

f(qz)

f(z)1113888 1113889

+ T rf(zq)

f(z)1113888 1113889 + O(log r)

T r g3(z)( 1113857 + T r g3z

q1113888 11138891113888 1113889 + O(log r)

2T r g3( 1113857 + S r g3( 1113857

2T rΔqf

f1113888 1113889 + S(r f)

(62)





f q2z1113872 1113873 + f(qz)

f(z) + f(zq)

d(qz)f(qz)

d(z)f(z) (64)


f(qz) g3(z)f(z)

f q2z1113872 1113873 g3(qz)f(qz)

g3(z)g3(qz)f(z)

fz

q1113888 1113889

f(z)

g3(zq)

(65)



f(z) + f(z)g3(zq)( 1113857

d(qz)

d(z)g3(z) (66)

that is

g3z

q1113888 1113889 g3(qz) + 1( 1113857

d(qz)

d(z)g

z

q1113888 1113889 + 11113888 1113889 (67)


N r g3z

q1113888 11138891113888 1113889 T r g3( 1113857 + S r g3( 1113857 (68)


(1 + o(1))N r g3( 1113857 N r g3z

q1113888 11138891113888 1113889

T r g3z

q1113888 11138891113888 1113889 + S r g3( 1113857

(1 + o(1))T r g3( 1113857 + S r g3( 1113857

(69)

is shows

N rΔqf

f1113888 1113889 N r g3( 1113857

T r g3( 1113857 + S r g3( 1113857

T rΔqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(70)


P5 z g3( 1113857 ≔ g3z

q1113888 1113889 g3(qz) + 1( 1113857

minusd(qz)

d(z)g

z

q1113888 1113889 + 11113888 1113889 equiv 0

(71)



P5(z ϱ + 1) (ϱ + 1)(ϱ + 1 + 1) minusd(qz)

d(z)(ϱ + 1 + 1)


d(z)1113888 1113889

(72)


m r1

Δqff1113872 1113873 minus ϱ⎛⎝ ⎞⎠ m r


1113888 1113889

S r g3( 1113857 S rΔqf

f1113888 1113889

(73)

is means

N r1

Δqff1113872 1113873 minus ϱ⎛⎝ ⎞⎠ T r

Δqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(74)



m r1

f(qz) + f(z)1113888 1113889 m r

f(z) + f(zq)

d(z)f(z)1113888 1113889

lem r1

d(z)1113888 1113889 + m r

f(z) + f(zq)

f(z)1113888 1113889

+ O(1)

le S(r f)

(75)


N r1

f(qz) + f(z)1113888 1113889 T r

1f(qz) + f(z)

1113888 1113889 + S(r f)

T r1

d(z)

f(z) + f(zq)

f(z)1113888 1113889 + S(r f)

T rf(zq)

f(z)1113888 1113889 + O(log r) + S(r f)

T rf(qz)

f(z)1113888 1113889 + S(r f)

T rΔqf

f1113888 1113889 + S(r f)

(76)




N r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ N r

1f(qz) + f(z)

1113888 1113889

T rΔqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(77)




m r1f

1113888 1113889lem r1

d(z)1113888 1113889 + m r

f(qz)

f(z)1113888 1113889 + m r

f(zq)

f(z)1113888 1113889

S(r f)

(78)

which implies

N r1f

1113888 1113889 T(r f) + S(r f) (79)


P6(z f) ≔ [f(qz) + f(z)] f(z) + fz


(80)



ϱ(4ϱ minus d(z))≢ 0(81)



N r1

f minus ϱ1113888 1113889 T(r f) + S(r f) (82)


m r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ S r

Δqf

f1113888 1113889

S(r f)

(83)



m r f(z) + fz

q1113888 11138891113888 1113889 m r

d(z)f(z)

f(qz) + f(z)1113888 1113889

m r1

(f(qz)f(z)) + 11113888 1113889 + O(log r)

m r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ + O(log r)

S(r f)

(84)



q1113888 11138891113888 1113889 + m(r f(qz) + f(z))

+ O(log r)

le 2m r f(z) + fz

q1113888 11138891113888 1113889 + m r

f(qz) + f(z)

f(z) + f(zq)1113888 1113889

+ O(log r)

S(r f)

(85)

which implies

N(r f) T(r f) + S(r f) (86)




q1113888 11138891113890 1113891 d(z)f(z)

(87)


N r1f

1113888 1113889leN r1Δqf(z)

1113888 1113889 + N r1

Δqf(zq)1113888 1113889 + N(r d)

2N r1Δqf

1113888 1113889 + S(r f)

(88)




f1113888 1113889 S(r f)

S rΔqf1113872 1113873

(89)

which implies


T r1Δqf

1113888 1113889 + S rΔqf1113872 1113873

(90)



P5(z z + 1) z

q+ 11113888 1113889(qz + 1 + 1) minus

d(qz)

d(z)

z

q+ 1 + 11113888 1113889

(z + q)(qz + 2)

qminus

d(qz)

d(z)

z + 2q

q

(91)


m r1

Δqff1113872 1113873 minus z⎛⎝ ⎞⎠ m r


1113888 1113889

S r g3( 1113857

S rΔqf

f1113888 1113889

(92)

is leads to

N r1

Δqff1113872 1113873 minus z⎛⎝ ⎞⎠ T r

Δqf

f1113888 1113889 + S r

Δqf

f1113888 1113889 (93)



Data Availability







Acknowledgments


References



































m rf(qz)

f(z)1113888 1113889 S(r f) (58)




m r Pq(z f)1113872 1113873 S(r f) (60)


f(qz)

f(z)+ 11113888 1113889

f(zq)

f(z)+ 11113888 1113889

d(z)

f(z) (61)


T(r f(z)) T rd(z)

f(z)1113888 1113889 + O(log r)leT r

f(qz)

f(z)1113888 1113889

+ T rf(zq)

f(z)1113888 1113889 + O(log r)

T r g3(z)( 1113857 + T r g3z

q1113888 11138891113888 1113889 + O(log r)

2T r g3( 1113857 + S r g3( 1113857

2T rΔqf

f1113888 1113889 + S(r f)

(62)





f q2z1113872 1113873 + f(qz)

f(z) + f(zq)

d(qz)f(qz)

d(z)f(z) (64)


f(qz) g3(z)f(z)

f q2z1113872 1113873 g3(qz)f(qz)

g3(z)g3(qz)f(z)

fz

q1113888 1113889

f(z)

g3(zq)

(65)



f(z) + f(z)g3(zq)( 1113857

d(qz)

d(z)g3(z) (66)

that is

g3z

q1113888 1113889 g3(qz) + 1( 1113857

d(qz)

d(z)g

z

q1113888 1113889 + 11113888 1113889 (67)


N r g3z

q1113888 11138891113888 1113889 T r g3( 1113857 + S r g3( 1113857 (68)


(1 + o(1))N r g3( 1113857 N r g3z

q1113888 11138891113888 1113889

T r g3z

q1113888 11138891113888 1113889 + S r g3( 1113857

(1 + o(1))T r g3( 1113857 + S r g3( 1113857

(69)

is shows

N rΔqf

f1113888 1113889 N r g3( 1113857

T r g3( 1113857 + S r g3( 1113857

T rΔqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(70)


P5 z g3( 1113857 ≔ g3z

q1113888 1113889 g3(qz) + 1( 1113857

minusd(qz)

d(z)g

z

q1113888 1113889 + 11113888 1113889 equiv 0

(71)



P5(z ϱ + 1) (ϱ + 1)(ϱ + 1 + 1) minusd(qz)

d(z)(ϱ + 1 + 1)


d(z)1113888 1113889

(72)


m r1

Δqff1113872 1113873 minus ϱ⎛⎝ ⎞⎠ m r


1113888 1113889

S r g3( 1113857 S rΔqf

f1113888 1113889

(73)

is means

N r1

Δqff1113872 1113873 minus ϱ⎛⎝ ⎞⎠ T r

Δqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(74)



m r1

f(qz) + f(z)1113888 1113889 m r

f(z) + f(zq)

d(z)f(z)1113888 1113889

lem r1

d(z)1113888 1113889 + m r

f(z) + f(zq)

f(z)1113888 1113889

+ O(1)

le S(r f)

(75)


N r1

f(qz) + f(z)1113888 1113889 T r

1f(qz) + f(z)

1113888 1113889 + S(r f)

T r1

d(z)

f(z) + f(zq)

f(z)1113888 1113889 + S(r f)

T rf(zq)

f(z)1113888 1113889 + O(log r) + S(r f)

T rf(qz)

f(z)1113888 1113889 + S(r f)

T rΔqf

f1113888 1113889 + S(r f)

(76)




N r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ N r

1f(qz) + f(z)

1113888 1113889

T rΔqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(77)




m r1f

1113888 1113889lem r1

d(z)1113888 1113889 + m r

f(qz)

f(z)1113888 1113889 + m r

f(zq)

f(z)1113888 1113889

S(r f)

(78)

which implies

N r1f

1113888 1113889 T(r f) + S(r f) (79)


P6(z f) ≔ [f(qz) + f(z)] f(z) + fz


(80)



ϱ(4ϱ minus d(z))≢ 0(81)



N r1

f minus ϱ1113888 1113889 T(r f) + S(r f) (82)


m r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ S r

Δqf

f1113888 1113889

S(r f)

(83)



m r f(z) + fz

q1113888 11138891113888 1113889 m r

d(z)f(z)

f(qz) + f(z)1113888 1113889

m r1

(f(qz)f(z)) + 11113888 1113889 + O(log r)

m r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ + O(log r)

S(r f)

(84)



q1113888 11138891113888 1113889 + m(r f(qz) + f(z))

+ O(log r)

le 2m r f(z) + fz

q1113888 11138891113888 1113889 + m r

f(qz) + f(z)

f(z) + f(zq)1113888 1113889

+ O(log r)

S(r f)

(85)

which implies

N(r f) T(r f) + S(r f) (86)




q1113888 11138891113890 1113891 d(z)f(z)

(87)


N r1f

1113888 1113889leN r1Δqf(z)

1113888 1113889 + N r1

Δqf(zq)1113888 1113889 + N(r d)

2N r1Δqf

1113888 1113889 + S(r f)

(88)




f1113888 1113889 S(r f)

S rΔqf1113872 1113873

(89)

which implies


T r1Δqf

1113888 1113889 + S rΔqf1113872 1113873

(90)



P5(z z + 1) z

q+ 11113888 1113889(qz + 1 + 1) minus

d(qz)

d(z)

z

q+ 1 + 11113888 1113889

(z + q)(qz + 2)

qminus

d(qz)

d(z)

z + 2q

q

(91)


m r1

Δqff1113872 1113873 minus z⎛⎝ ⎞⎠ m r


1113888 1113889

S r g3( 1113857

S rΔqf

f1113888 1113889

(92)

is leads to

N r1

Δqff1113872 1113873 minus z⎛⎝ ⎞⎠ T r

Δqf

f1113888 1113889 + S r

Δqf

f1113888 1113889 (93)



Data Availability







Acknowledgments


References

































P5(z ϱ + 1) (ϱ + 1)(ϱ + 1 + 1) minusd(qz)

d(z)(ϱ + 1 + 1)


d(z)1113888 1113889

(72)


m r1

Δqff1113872 1113873 minus ϱ⎛⎝ ⎞⎠ m r


1113888 1113889

S r g3( 1113857 S rΔqf

f1113888 1113889

(73)

is means

N r1

Δqff1113872 1113873 minus ϱ⎛⎝ ⎞⎠ T r

Δqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(74)



m r1

f(qz) + f(z)1113888 1113889 m r

f(z) + f(zq)

d(z)f(z)1113888 1113889

lem r1

d(z)1113888 1113889 + m r

f(z) + f(zq)

f(z)1113888 1113889

+ O(1)

le S(r f)

(75)


N r1

f(qz) + f(z)1113888 1113889 T r

1f(qz) + f(z)

1113888 1113889 + S(r f)

T r1

d(z)

f(z) + f(zq)

f(z)1113888 1113889 + S(r f)

T rf(zq)

f(z)1113888 1113889 + O(log r) + S(r f)

T rf(qz)

f(z)1113888 1113889 + S(r f)

T rΔqf

f1113888 1113889 + S(r f)

(76)




N r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ N r

1f(qz) + f(z)

1113888 1113889

T rΔqf

f1113888 1113889 + S r

Δqf

f1113888 1113889

(77)




m r1f

1113888 1113889lem r1

d(z)1113888 1113889 + m r

f(qz)

f(z)1113888 1113889 + m r

f(zq)

f(z)1113888 1113889

S(r f)

(78)

which implies

N r1f

1113888 1113889 T(r f) + S(r f) (79)


P6(z f) ≔ [f(qz) + f(z)] f(z) + fz


(80)



ϱ(4ϱ minus d(z))≢ 0(81)



N r1

f minus ϱ1113888 1113889 T(r f) + S(r f) (82)


m r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ S r

Δqf

f1113888 1113889

S(r f)

(83)



m r f(z) + fz

q1113888 11138891113888 1113889 m r

d(z)f(z)

f(qz) + f(z)1113888 1113889

m r1

(f(qz)f(z)) + 11113888 1113889 + O(log r)

m r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ + O(log r)

S(r f)

(84)



q1113888 11138891113888 1113889 + m(r f(qz) + f(z))

+ O(log r)

le 2m r f(z) + fz

q1113888 11138891113888 1113889 + m r

f(qz) + f(z)

f(z) + f(zq)1113888 1113889

+ O(log r)

S(r f)

(85)

which implies

N(r f) T(r f) + S(r f) (86)




q1113888 11138891113890 1113891 d(z)f(z)

(87)


N r1f

1113888 1113889leN r1Δqf(z)

1113888 1113889 + N r1

Δqf(zq)1113888 1113889 + N(r d)

2N r1Δqf

1113888 1113889 + S(r f)

(88)




f1113888 1113889 S(r f)

S rΔqf1113872 1113873

(89)

which implies


T r1Δqf

1113888 1113889 + S rΔqf1113872 1113873

(90)



P5(z z + 1) z

q+ 11113888 1113889(qz + 1 + 1) minus

d(qz)

d(z)

z

q+ 1 + 11113888 1113889

(z + q)(qz + 2)

qminus

d(qz)

d(z)

z + 2q

q

(91)


m r1

Δqff1113872 1113873 minus z⎛⎝ ⎞⎠ m r


1113888 1113889

S r g3( 1113857

S rΔqf

f1113888 1113889

(92)

is leads to

N r1

Δqff1113872 1113873 minus z⎛⎝ ⎞⎠ T r

Δqf

f1113888 1113889 + S r

Δqf

f1113888 1113889 (93)



Data Availability







Acknowledgments


References

































m r f(z) + fz

q1113888 11138891113888 1113889 m r

d(z)f(z)

f(qz) + f(z)1113888 1113889

m r1

(f(qz)f(z)) + 11113888 1113889 + O(log r)

m r1

Δqff1113872 1113873 + 2⎛⎝ ⎞⎠ + O(log r)

S(r f)

(84)



q1113888 11138891113888 1113889 + m(r f(qz) + f(z))

+ O(log r)

le 2m r f(z) + fz

q1113888 11138891113888 1113889 + m r

f(qz) + f(z)

f(z) + f(zq)1113888 1113889

+ O(log r)

S(r f)

(85)

which implies

N(r f) T(r f) + S(r f) (86)




q1113888 11138891113890 1113891 d(z)f(z)

(87)


N r1f

1113888 1113889leN r1Δqf(z)

1113888 1113889 + N r1

Δqf(zq)1113888 1113889 + N(r d)

2N r1Δqf

1113888 1113889 + S(r f)

(88)




f1113888 1113889 S(r f)

S rΔqf1113872 1113873

(89)

which implies


T r1Δqf

1113888 1113889 + S rΔqf1113872 1113873

(90)



P5(z z + 1) z

q+ 11113888 1113889(qz + 1 + 1) minus

d(qz)

d(z)

z

q+ 1 + 11113888 1113889

(z + q)(qz + 2)

qminus

d(qz)

d(z)

z + 2q

q

(91)


m r1

Δqff1113872 1113873 minus z⎛⎝ ⎞⎠ m r


1113888 1113889

S r g3( 1113857

S rΔqf

f1113888 1113889

(92)

is leads to

N r1

Δqff1113872 1113873 minus z⎛⎝ ⎞⎠ T r

Δqf

f1113888 1113889 + S r

Δqf

f1113888 1113889 (93)



Data Availability







Acknowledgments


References



































Acknowledgments


References

































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ResultsonSolutionsforSeveral -Painleve´Difference ...downloads.hindawi.com/journals/jmath/2020/3781942.pdf · (iii)If ξ 0 and Y(z) −Ahas ﬁnite zeros, then Ais a solutionof3z2