α-normal functions and yosida functions

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α-normal functions and yosida functionsShamil Makhmutov aa Department of Mathematics , Ufa State Aviation Technical University , Ufa, RussiaPublished online: 26 Jun 2007.

To cite this article: Shamil Makhmutov (2001) α-normal functions and yosida functions, Complex Variables, Theory andApplication: An International Journal: An International Journal, 43:3-4, 351-362

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a-normal Functions and Yosida Functions

SHAMlL MAKHMUTOV*

Department of Mathematics, Ufa State Aviation Technical University, Ufa, Russia

Communicated by R. P. Gilbert

(Received 3 January 2000)

a-normal functions, a 2 1, are meromorphic functions in the unit disk D with

a-normal functions ( a > 1) are characterized by the normality of a family of functions { fn(C)}, fn(c) = f(an+(l - lan12)u~), limn,, la,l= 1. on compact subsets of the finite complex plane C. We prove that limit functions of converging sequences of functions {fn(<)) are Yosida's functions. We extract a subclass of the a-normal functions such that limit functions of converging sequences of functions are Yosida's functions of the first kind. Note that meromorphic solutions of algebraic differential equations of the &st order with coefficients from the Hardy spaces HP are a-normal functions.

Keywordr and Phrases: a-normal function; Yosida function

1991 Mathematics Subject Class~jications: Primary: 30D35; Secondary: 30D50, 30D45

In [4] we obtained the estimate of the growth of spherical derivative of meromorphic solutions of differential equations of the first order with coefficients from Hardy spaces HP in the unit disk

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D= { z E C: JzJ < 1 ) . Meromorphic solution w of the differential equation

and akj(z) E H P ~ J , satisfies the condition

where a = maxl 5 k 5 , maxo 5,s ,, ( 1 / (kpk,)) and w#(z) = 1 wl(z)((l + 1 w(z)12) - ' is the spherical derivative of w(z).

DEFINITION 1 A function f meromorphic in D is called a-normal ( a > 0) in D if it satisfies the condition (2). The set of these functions is denoted by N,.

In [ l , 2,4] a-normal meromorphic functions (a 2 1 ) are described in terms of Pa-sequences of points, mean values of integrals involving spherical derivative. In the case of 0 < a < 1 Yamashita [6] proved that the class N , coincides with the spherical Lipschitz class.

In this paper we will study a-normal meromorphic functions from the position of normality.

Consider the case a = 1 . It is known [3,5] that f E N = N1, i.e., f is a normal function in D, if and only if the family of functions { f((z+a)/( l +az))} , a € D, is normal in the unit disk D. If addi- tionally there is no converging sequence of functions { f ( ( z f a,)/ ( 1 +ii,z))}, lim,,, )an) = 1, which has a constant as a limit function, then f is called normal function of the first kind [5 ] . We consider the convergence of sequences of meromorphic functions in the chordal metric x ( . ,.). In the case of normal functions limit functions of converging sequences { f ((z + a,)/(l + ii,z))) are also normal functions. Iff is a normal function of the first kind then limit functions of corresponding converging sequences are also normal functions of the first kind (see [5] ) .

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YOSIDA FUNCTIONS 353

Yosida [5] introduced the class (A) of meromorphic functions on the complex plane C. By the definition, a function f belongs to the class (A) if the family of functions { f (z+a)}, a~ C , is normal in C and there are no sequences {a,}, limn,, lan[ = m, such that the sequence of functions { f(z+an)} converges to a constant then f is called Yosida function of the first kind. For example, any elliptic function on C is a Yosida function of the first kind [7].

THEOREM A An Yosida function f is of the jirst kind if and only if for any 6 > 0 there exists r > 0 such that

Next theorem describes the value distribution property of Yosida functions of the first kind.

THEOREM B For every Yosida function of thejirst kind g(z) there is a positive number Rg such that g(z) takes every value of the extended complex plane in every disk Iz - a1 I Rg, a E C.

1. LIMIT PROPERTIES OF a-NORMAL FUNCTIONS

Let a > 1. Gavrilov [I] proved (see also [2] ) that a meromorphic function f in D is an a-normal function in D if and only if for any sequence of points {a,) c D, limn,, lanl = 1 , the family of functions { f (an+( l - lanl)"z)) is normal on compact subsets of the complex plane C.

Let K R = { z ~ C : I z I SR}, O < R < w .

LEMMA 1 For any R > 0 there is a number 6, 0 < 6 < 1 , such that

for any a € ( 1 -6 < la1 < I } .

Condition (3) is equivalent to

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for any < E KR. In fact,

If we take 6 satisfying the condition 0 < 6 * - ' ~ < 1 , then (3 ) is valid. We say that function f is an a-normal function of the first kind if

there is no sequence {a,} c D such that the sequence of functions { f (a, + ( 1 - la,[ )*z)} converges to constant on compact subsets of the unit disk D.

LEMMA 2 Let {a,} c D, limn,, la,l= 1 , and w, = a,+(l - (a,l )"z, where z E KR and R is a fixed positive number. Then

In fact,

Since a > 1 and lanl + 1 as n + oo, condition (4) is realized.

THEOREM 1 For any f E N,, a > 1, and any sequence {a,} c D, limn,, la,l = 1 , non-constant limit functions of convergent sequences of functions { f (a,+(l - la,[ )"z)} are Yosida functions.

Proof Let w, = a, + (1 - la,l )%. By Lemma 1 , for any fixed R > 0 there exists a number N such that beginning with n > N points w, are in D for any z E KR. Without loss of generality we assume that sequence { f(a,+(l - la,l)*z)} converges uniformly to non-constant function g(z) on compact subsets of C. We estimate the growth of spherical derivative of g(z).

g # ( ~ ) = lim f,#(z) = lim ( 1 - ~z,l)"f#(w,) ,403 n-403

= lim ( I - Iw,l)"f#(w,) < m. n+oo

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Thus, g#(z) is bounded on compact subsets of C and therefore g(z) is a Yosida function.

Now we give a necessary and sufficient condition for an a-normal function to be of the first kind.

Denote D,(a, 6) = { z E D: lz- a1 I ( 1 - la1 )"6)) and D(a, 6) = Do (a, 4. LEMMA 3 Let f E N,, a > 1 . Then f is of the jirst kind if and only if for every E > 0 there is an 6 > 0 such that

lim inf rnax ( 1 - l ~ l ) ~ f # ( z ) > E . 1~1-1 z ED,(a,6)

Proof Necessity Assume that ( 5 ) is not realized. Th sequence {a,) c D such that for some 6 > 0

.en there exists a

lim inf rnax ( 1 - lzl)"f#(z) = 0. lan/+l zED,(an,6)

Consider the sequence of functions { f,(z)), fn(z) = f (a, + ( 1 - lan[ )"z). Without loss of generality we assume that this sequence of functions converges uniformly to a non-constant function g(z) in the neighbourhood of origin. Assuming w, = a, + ( 1 - lanl )@z and by Lemma 2

rnax g# (z) = lim rnax f,# ( z ) 121 5 6 n+m 121 5 6

= lim max(1 - Iw,(z)l)*f#(w,(z)) n + w 121 5 6

= lim rnax ( 1 - lzl)"f#(z) = 0. n+" z E D,(a,,6)

Since g(z) is a non-constant function we obtain a contradiction.

Necessity Assume that f is not an a-normal function of the first kind. Then there exists a sequence {a,) c D, limn,, (a,J = 1 , such that the sequence of functions { f(a,+(l - la,l)"z)} converges uniformly to a constant in a neighbourhood of the origin. By (6) we obtain a contradiction with (5).

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2. CERCLES DE REMPLISSAGE

In Theorem 1 we proved that iff is an a-normal function on the unit disk D then limit functions of converging sequences of functions { f (an+ ( 1 - (a,l )az)}, limn,, lanl = 1, are Yosida functions.

We next consider the case when f is an a-normal function of the first kind.

THEOREM 2 Iff is an a-normal function of thefirst kind, a > 1, then limit function of converging sequences { f (a, + ( 1 - lanl )*z)} are Yosida functions of the first kind.

Proof We know that the family of functions { f (an+ ( 1 - lanl )"z)} is normal on C. Without loss of generality we assume that this sequence converges to the Yosida function 4(z). Suppose that 4(z) is not of the first kind. Then by Lemma A there is a sequence {ck}, limk.+, lckl = co, on which

lim max $#(z + ck) = 0 k-rm 121 5 6

(7)

for each fixed 6 > 0. For every number k one has

Let &(z) = an+ ( 1 - la,l )" ck+(l- lan[ )az and bnk(z) =an+ ( I - lan[ ) a ~ k . Then

max ##(z + ck) = lim max(1 - lanl)*f#(~nk(z)). 121 5 6 n-., 121 5 6

Note that for any number k

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Thus, we have

For any fixed 6 > 0 and any z satisfying Iz1 I 6, if Cnk = bnk+ ( 1 - lbnkl)a~, then

Since a > 1 and by (9) we obtain

In the same way we can show that

Thus, applying (9), (lo), (11) to (8) and using uniform convergence on argument z and k we obtain

= lirn { man (1 - lzl)CIf#(z)} "'00 z € D, (b&,6)

2 lim inf{ max (1 - jrl)yx(z)) = ~ ( 6 ) > O . lal-1 z € D,(a,b)

The last contradicts assumption (7).

The next result is a consequence of Theorems B and 2.

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THEOREM 3 For every a-normal function f of the first kind, cu > 1, there is a number Rf > 0 such that f (z) takes in every disk D,(a, Rf ), 0 < ro 5 la1 < 1, every value of the extended complex plane C.

Condition la1 1 ro is necessary only for the inclusion D,(a, Rf) c D.

Proof To prove the theorem by contradiction, we assume that there are sequences {a,} c D, limn,, lan/ = 1, and {r,}, 0 < r, 5 r,, where limn,, r, = oo, such that all disks D,(a,, r,) c D. We assume that there is a sequence {b,) c such that the function f does not take the value b, in the disk D,(a,, r,) for each n. Without loss of generality we set that limn,, b, = bo EC and that the sequence of functions { fn(z)>, f,(z) = f (an+(l - lanl ),z), converges to a function g(z) uniformly on compact subsets of C. By Theorem 2 the function g(z) is a Yosida function of the first kind. By Theorem B there is a number Rg > 0 such that g(z) takes every value in in every disk D(a, R,), UEC.

Take such a point zo E D(0, R,) that g(zo) = bo. For any E > 0 there exists a 6 > 0 such that the inequality x(g(z), bo) < (44) is valid in the disk D(zo,6) and X ( f,(z), bo) < ( 4 2 ) is valid in the disk D(zo,6) for sufficiently big n too. The last is equivalent to

where zEQ(n,zo,6)={z~D:z=an+(l-~a,~)azo+(1-~a,~)"~,~~~~ 6). The inclusion Q(n, zo, 6) c D holds for any fixed 6 > 0 and any sufficiently big n, i.e., n > N(6). In fact,

We notice that if limn,, Ja,J = 1, a > 1, I[) 5 6, )zol 5 Rg, then

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YOSIDA FUNCTIONS

for n such that (1 - la,[ )"-'Izo + < 1. Since

lim an + (1 - Janl)az~ - n+oa - 1,

a,

we can choose a number N2 such that D,(a,, (612)) c Q(n, zo, 6) for n 2 N2. Rewrite condition limn,, ~ (b , , bo) = 0 as ~ (b , , bo) 5 (44) for n 2 N3. By the triangle inequality we obtain that

holds for z E Da(a,, (612)) and n 2 N = max{N1, N2, N3}. The last contradicts the assumption that f(z) does not take the value b, in the disk D,(a,, r,), n > N, since the assumption D,(a,, (612)) c D,(a,, r,) for every n such that r, > (612).

To prove our next theorem we need the following definition.

DEFINITION [l] A sequence {z,} c D, limn,, lznl = 1 is called a Pa- sequence, a > l , of a meromorphic function f (z) if for each 6 > 0 and each subsequence {z,,) the function f(z) takes in UEl D,(z,,, 6) all values from the extended complex plane -d infinitely often with the possible exception of two values.

Gavrilov [ I ] proved that f is an a-normal function if and only if it doesn't possess Pa-sequences of points.

For our convenience we denote

In ([4], Lemma 2.5) it was proved that if {a,} and {b,} are sequences in D and limn,, la,l = 1 with lirn,,, lbnl = I , then the conditions

inf P(a,, bm) > 0 n,m

inf P(bm, a,) > 0 n.m

are equivalent. For simplicity of notation we denote the a-points of the function

f as z,(a) or z,(f, a) where it will be needed. Dow

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THEOREM 4 A meromorphic function f ( z ) in D is an a-normal function of the first kind, a > 1 , if and only if it satisfies the following conditions:

( A ) there is an R > 0 such that at least one zero and one pole of f ( z ) belong to the disk D,(a, R) , 0 i ro 5 la/ < 1 ;

(B) infv,p(lz,@,f 1-zw(O,f )1)/((1 - I ~ w ( O 7 f ) I Y)) > 0; (C) for any r, 0 < r < 1, there exists q, 0 < q < 1, such that

X ( f (4, m) > 9 for every z E D\ u;=, Da (2, (m,f 1, r) and x ( f (470) > qfor every z E D\ UZl D&U(O,f 1 1 r).

The condition 0 5 ro 5 la1 < 1 in (A) is necessary for the inclusion D,(a, R ) c D.

Proof (A) is proved in Theorem 3 and (B) is proved in [4]. Since the functions f ( z ) and ((af (z)+ b)/(cf(z) + d )), ad- bc # 0, are

a-normal functions of the first kind at the same time it is enough to consider condition ( C ) for arbitrary a € C , i.e., to show that x ( f ( z ) , a) > q for z E D\ UF1 Da(zj(a), r).

Let a = 0. We assume that

inf x ( f ( z ) , 0 ) = 0 z E B, ( 12)

where B, = D\ Uz, D, ( z , ( 0 ) , r). Then there exists a sequence {a,} c B, such that limn,, f (a,) = 0. Set fn(z) = f (a, + ( 1 - la,[ ),z), n = 1,2, . . . . Since f is an a-normal function of the first kind we can assume without loss of generality that the sequence f,(z) converges uniformly to the non-constant function g(z).

g(0) = lim f,(O) = lim f(a,) = 0. n+w n + w

Take 6, 0 < 6 < 1 , so small that the limit function g(z) has only one zero in the disk lzl < 6. It is possible to extract such a disk since g(z) is a Yosida function of the first kind [7]. Let m be the order of the zero. By the Hurwitz theorem the number of zeros of f,(z) in the disk {z: I z I 5 6 ) is also m for sufficiently big n. It means that the number of zeros of f ( z ) in each disk D,(an, 6) is also m for the same n. On the other hand, since {a,} c B, and assumption (12) we can find such a small 6 > 0 that there are not disks D,(an, 6) containing zeros {zw(0)} for all sufficiently big n. This contradicts (12).

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For the proof of sufficiency we assume that f is not an a-normal function of the first kind. Then there are two possibilities:

- f is an a-normal function of the second kind, - f is not an a-normal function.

The first possibility is excluded since by (A) limit functions of converging sequences { f(a,+ (1 - lan[ )"z)} are not constant.

Consider the second possibility. If f is not a-normal function then there is a Pa-sequence {a,} of f(z) (see [I]). Two cases are possible:

In case (a) we take 6 such that

According to (C) the function f is bounded on IJ,"=, Da(an, 6) in the sense of the chordal metric. The last fact contradicts our assumption that {a,} is Pa-sequence of f(z).

In case (b) we take a subsequence {a,,) c {a,) and {z,(O)) c {z,(O)) on which

According to (B) and Lemma 2.5 of [4]

Take 61 such that

P(a,, z, (co)) , inf p' (a,, zp (00)) w D

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S. MAKHMUTOV

By (C) the function f is bounded on (JEl D,(an, 61). It contradicts the assumption that {a,} is a Pa-sequence of f ( z ) .

COROLLARY I f f are g are a-normal functions of theJirst kind and the poles and zeros of these functions are connected by

inf Iz,(m, g) - z @ , f I > 0 , V I P (1 - l ~ P ( 0 W ) l ) ~

inf Izp(m,f) - s ( 0 , g ) l > 0 , Y , , ( l - I ~ P ( m , f ) l ) ~

then the product f . g is an a-normal function of the first kind.

References

[I] Gavrilov, V. 1. (1965). On the value distribution of functions meromorphic in the unit circle which are not normal, Mat. Sbornik, 67, 408-427 (Russian).

[2] Kanatnikov, A. N. (1982). Two criteria for PA-sequences, Bulletin of Moscow University, No. 3, pp. 33-35.

[3] Lehto, 0, and Virtanen, K. I. (1957). Boundary behaviour and normal meromorphic functions, Acta Math., 97, 47- 65.

[4] Makhmutov, S. (1998). The distribution of a-points of meromorphic functions and the growth of spherical derivative, Bulletin of the Hong Kong Math. Soc., 2, 89-98.

[5] Noshiro, K. (1938). Contributions to the theory of meromorphic functions in the unit circle, J. Fac. Sci., Hokkaido Univ., 7, 149- 159.

[6] Yamashita, Sh. (1988). Meromorphic Lipschitz functions, Bull. Austral. Math. Soc., 38(3), 451 -455.

[7] Yosida, K. (1934). On a class (A) of meromorphic functions, Proc. Phys.-Math. Soc. Japan, 16, 227-235.

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α-normal functions and yosida functions