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Volume 3, Number 3, 2015

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Feng Qi Henan Polytechnic University, China

Cenap Özel Dokuz Eylül University, Turkey

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Erdoğan Şen Namik Kemal University, Turkey

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M. E. H. Ismail University of Central Florida, United States

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Editors

Henry W. Gould West Virginia University, United States

Toka Diagana Howard University, United States

Abdelmejid Bayad Université d'éry Val d'Essonne, France

Hassan Jolany Université de Lille 1, France

István Mező Nanjing University of Information Science and Technology, China

C. S. Ryoo Hannam University, South Korea

Junesang Choi Dongguk University, South Korea

Dae San Kim Sogang University, South Korea

Taekyun Kim Kwangwoon University, South Korea

Guotao Wang Shanxi Normal University, China

Yuan He Kunming University of Science and Technology, China

Aleksandar Ivıc Katedra Matematike RGF-A Universiteta U Beogradu, Serbia

Cristinel Mortici Valahia University of Targoviste, Romania

Naim Çağman University of Gaziosmanpasa, Turkey

Ünal Ufuktepe Izmir University of Economics, Turkey

Cemil Tunc Yuzuncu Yil University, Turkey

Abdullah Özbekler Atilim University, Turkey

Donal O'Regan National University of Ireland, Ireland

S. A. Mohiuddine King Abdulaziz University, Saudi Arabia

Dumitru Baleanu Çankaya University, Turkey

Ahmet Sinan CEVIK Selcuk University, Turkey

Erol Yılmaz Abant Izzet Baysal University, Turkey

Hünkar Kayhan Abant Izzet Baysal University, Turkey

Yasar Sozen Hacettepe University, Turkey

I. Naci Cangul Uludag University, Turkey

İlkay Arslan Güven University of Gaziantep, Turkey

Semra Kaya Nurkan University of Uşak, Turkey

Ayhan Esi Adiyaman University, Turkey

M. Tamer Kosan Gebze Institute of Technology, Turkey

Hanifa Zekraoui Oum-El-Bouaghi University, Algeria

Siraj Uddin University of Malaya, Malaysia

Rabha W. Ibrahim University of Malaya, Malaysia

Adem Kilicman University Putra Malaysia, Malaysia

Armen Bagdasaryan Russian Academy of Sciences, Moscow, Russia

Viorica Mariela Ungureanu University Constantin Brancusi, Romania

Valentina Emilia Balas “Aurel Vlaicu” University of Arad, Romania

R.K Raina M.P. Univ. of Agriculture and Technology, India

M. Mursaleen Aligarh Muslim University, India

Vijay Gupta Netaji Subhas Institute of Technology, India

Hemen Dutta Gauhati University, India

Akbar Azam COMSATS Institute of Information Technology, Pakistan

Moiz-ud-din Khan COMSATS Institute of Information Technology, Pakistan

Roberto B. Corcino Cebu Normal University, Philippines

Turkish Journal of Analysis and Number Theory, 2015, Vol. 3, No. 3, 75-77

Available online at http://pubs.sciepub.com/tjantr/3/3/1

© Science and Education Publishing

DOI:10.12691/tjant-3-3-1

Backward Orbit Conjecture for Lattès Maps

Vijay Sookdeo*

Department of Mathematics, The Catholic University of America, Washington, DC

*Corresponding author: [email protected]

Received March 05, 2015; Accepted May 16, 2015

Abstract For a Lattès map 1 1: defined over a number field K, we prove a conjecture on the integrality

of points in the backward orbit of ( )P K under .

Keywords: backward orbit conjecture, Lattès maps

Cite This Article: Vijay Sookdeo, “Backward Orbit Conjecture for Lattès Maps.” Turkish Journal of

Analysis and Number Theory, vol. 3, no. 3 (2015): 75-77. doi: 10.12691/tjant-3-3-1.

1. Introduction

Let 1 1: P be a rational map of degree 2

defined over a number field ,K and write n for the

nth iterate of . For a point 1,P let

2, , ,...P P P P be the forward orbit of P

under , and let

0

n

n

P P

be the backward orbit of P under . We say P is

-preperiodic if and only if P is finite.

Viewing the projective line 1 as 1 and

taking 1 ,P K a theorem of Silverman [4] states

that if is not a fixed point for 2 , then P

contains at most finitely many points in ,K the ring of

algebraic integers in .K If S is the set of all

archimedean places for ,K then K is the set of points

in 1 K which are S-integral relative to (see section

2). Replacing with any point 1Q K and S with

any finite set of places containing all the archimedean

places, Silverman's Theorem can be stated as: If Q is not

a fixed point for 2 , then P contains at most

finitely many points which are S-integral relative to .Q

A conjecture for finiteness of integral points in

backward orbits was stated in [[6], Conj. 1.2].

Conjecture 1.1. If 1( )Q K is not S-preperiodic, then

P contains at most finitely many points in 1( )K

which are S-integral relative to .Q

In [6], Conjecture 1.1 was shown true for the powering

map dz z with degree 2,d and consequently

for Chebyschev polynomials. A gener-alized version of

this conjecture, which is stated over a dynamical family of

maps , is given in [[1], Sec. 4]. Along those lines, our

goal is to prove a general form of Conjecture 1.1 where

is the family of Lattès maps associate to a fixed

elliptic curve E defined over K (see Section 3).

2. The Chordal Metric and Integrality

2.1. The Chordal Metric on .N Let KM be the set

of places on K normalized so that the product formula

holds: for all *,K

1.v

v MK

For points 0 1: : : NP x x x and

0 1: : : NQ y y y in ( ),NvK define the v-adic

chordal metric as

,max

, .max .max

i j i j j i vv

i i i iv v

x y x yP Q

x y

Note that v is independent of choice of projective

coordinates for P and Q, and 0 , 1v (see [2]).

2.2. Integrality on Projective Curves. Let C be an

irreducible curve in N defined over K and S a finite

subset of KM which includes all the archimedean places.

A divisor on C defined over K is a finite formal sum

i in Q with in and .iQ C K The divisor is

76 Turkish Journal of Analysis and Number Theory

effective if 0in for each i, and its support is the set

Supp(D) = 1, , .Q Ql

Let , log ,Q v vP P Q and

, ,D v i Q viP n P when .i iD n Q This makes

,D v an arithmetic distance function on C (see [3]) and

as with any arithmetic distance function, we may use it to

classify the integral points on C.

For an effective divisor i iD n Q on C defined over

K , we say P C K is S-integral relative to D, or P is a

(D, S)-integral point, if and only if ,

0Q vi

P for

all embeddings , : K K and for all places .v S

Furthermore, we say the set C K is S-integral

relative to D if and only if each point in is S-integral

relative to D.

As an example, let C be the projective line 1 ,

S be the Archimedean place of ,K and .D

For / ,P x y with x and y are relatively prime in ,

we have , logD v vP y for each prime v. Therefore,

P is S-integral relative to D if and only if 1;y that is,

P is S-integral relative to D is and only if .P

From the definition we find that if 1 2S S are finite

subsets of KM which contains all the archimedean

places, then P is a 2,D S -integral point implies that P is

a 1,D S -integral point. Similarly, if Supp 1D

Supp 2D , then P is a 2 ,D S -integral point implies that

P is also a 2 ,D S -integral point. Therefore enlarging S

or Supp(D) only enlarges the set of ,D S -integrals points

on .C K

For 1 2: C C a finite morphism between projective

curves and 2 ,P C write

*

1Q P

P e Q Q

where 1e Q is the ramification index of at Q.

Furthermore, if i iD n Q is a divisor on C, then we

define * * .i iD n Q

Theorem 2.1 (Distribution Relation). Let 1 2: C C be

a finite mor-phism between irreducibly smooth curves in

( ).N K Then for 1,Q C there is a finite set of places S,

depending only on and containing all the archimedean

places, such that , * ,P v P v for all .v S

Proof. See [[3], Prop. 6.2b] and note that for projective

varieties the W V term is not required, and that the

big-O constant is an KM -bounded constant not

depending on P and Q.

Corollary 2.2. Let 1 2: C C be a finite morphism

between irreducibly smooth curves in ( ),N K let

1( ),P C K and let D be an effective divisor on C2 defined

over K. Then there is a finite set of places S, depending

only on and containing all the archimedean places,

such that P is S-integral relative to D if and only P is

S-integral relative to * .D

Proof. Extend S so that the conclusion of Theorem 2.1

holds. Then for i iD n Q with each 0in and

2 ( ),iQ C K we have that.

* , ,,.D v i Q viD v

P P n P

So * ,0

D vP

if and only if , 0.Q vi

P

3. Main Result

Let E be an elliptic curve, : E E a morphism, and

1: E be a finite covering. A Lattès map is a

rational map 1 1: making the following diagram

commute:

For instance, if E is defined by the Weierstrass

equation 2 3 2 ,y x ax bx c [2] is the

multiplication-by-2 endomorphism on E, and

, ,x y x then

4 2 2

3 2

2 8 4.

4 4 4 4

x bx cx b acx

x ax bx c

Fix an elliptic curve E defined over a number field K,

and for 1( )P K define:

1 1 1

:

: cov :

there exist K morphosm E E

and finite ering E such

that

0 P

1

0 ( ) preperK

A point Q is -preperiodic if and only if Q is -

preperiodic for some . We write 1( ) preperK

for the set of -preperiodic points in 1( ).K

Turkish Journal of Analysis and Number Theory 77

Theorem 3.1. If 1( )Q K is not -periodic, then

contains at most finitely many points in 1( )K which are

S-integral relative to Q.

Proof. Let 0 be the End(E)-submodule of ( )E K that

is finitely generated by the points in 1 ,P and let

0( ) for some non-zero End .E K E

Then 1 . Indeed, if is not -

preperiodic, then is non torsion and 1 0

for some Lattès map 1 . So 1 0 for some

morphism 1 : ,E E and this gives 1 2 P

for some Lattès map 2 . Therefore 11 2 P

12 P for some morphism 2 : .E E Since

any morphism : E E is of the form

X X T where End E and torsT E

(see [[5], 6.19]), we find that there is a End E such

that is in 0 , the End(E)-submodule generated by

1 .P Otherwise, if is -preperiodic, then

1( ) prepertors

E K K ([[5], Prop. 6.44]) gives

that may be a torsion point; again since

.tors

E K Hence 1 .

Let D be an effective divisor whose support lies entirely

in 1 ,Q let Q be the set of points in which are

S-integral relative to Q, and let D be the set of points in

which are S-integral relative to D. Extending S so that

Theorem 2.1 holds for the map 1: E , and since

Supp(D) Supp *D , we have: if is S-integral

relative to Q, then 1 is S-integral relative to D.

Therefore 1Q D . Now is a finite map and

1( );E K K so to complete the proof, it suffices

to show that D can be chosen so that D is finite.

From [[5], Prop. 6.37], we find that if is a nontrivial

subgroup of Aut(E), then 1E and the map

: E can be determine explicitly. The four

possibilities for , which are 2 3, , , ,x y x x x or y

correspond respectively to the four possibilities for ,

which are 2 4 6, , , or 3 , which in turn depends

only on the j-invariant of E. (Here, N denotes the Nth

roots of unity in .)

First assume that ,x y y . Since Q is not [']-

preperiodic, take 1 Q to be non torsion. Then

1 Q since 2 4 6, , ,or and

2 is non-torsion. Taking ,D

[[1], Thm. 3.9(i)] gives that D is finite.

Suppose that , .x y y Then , , ,x y

where 0 and is non-torsion since Q is

not -preperiodic. Assuming that both and

are torsion give that 3 is torsion, and this

contradicts the fact that is torsion. Therefore, we may

assume that is non-torsion. Now taking

,D [[1], Thm. 3.9(i)] again gives that D is

finite. Hence RQ, the set of points in which are S-

integral relative to Q, is finite.

References

[1] David Grant and Su-Ion Ih, Integral division points on curves,

Compositio Math-ematica 149 (2013), no. 12, 2011-2035.

[2] Shu Kawaguchi and J. H. Silverman, Nonarchimedean green

functions and dynam-ics on projective space, Mathematische

Zeitschrift 262 (2009), no. 1, 173-197.

[3] J. H. Silverman, Arithmetic distance functions and height

functions in Diophantine geometry, Mathematische Annalen 279

(1987), no. 2, 193-216.

[4] Integer points, Diophantine approximation, and iteration of

rational maps, Duke Math. J. 71 (1993), no. 3, 793-829.

[5] The arithmetic of dynamical systems, Graduate Text in

Mathematics 241, Springer, New York, 2007.

[6] V. A. Sookdeo, Integer points in backward orbits, J. Number

Theory 131 (2011), no. 7, 1229-1239.


Available online at http://pubs.sciepub.com/tjant/3/3/2


DOI:10.12691/tjant-3-3-2

Occasionally Weakly Compatible Mappings

Amit Kumar Govery1,*

, Mamta Singh2

1School of Studies in Mathematics, Vikram University, Ujjain - 456010 (M.P.), India 2Department of Mathematical Science and Computer Application, Bundelkhand University, Jhansi (U.P.), India



Abstract In this paper, the concept of compatible maps of type (A) and occasionally weakly compatible maps in

fuzzy metric space have been applied to prove common fixed point theorem. A fixed point theorem for six self maps

has been established using the concept of compatible maps of type (A) and occasionally weakly compatible maps,

which generalizes the result of Cho [16].

Keywords: common fixed points, fuzzy metric space, compatible maps, compatible maps of type (A) and

occasionally weakly compatible maps

Cite This Article: Amit Kumar Govery, and Mamta Singh, “Occasionally Weakly Compatible Mappings.”

Turkish Journal of Analysis and Number Theory, vol. 3, no. 3 (2015): 78-82. doi: 10.12691/tjant-3-3-2.

1. Introduction

The concept of Fuzzy sets was initially investigated by

Zadeh [13] as a new way to represent vagueness in

everyday life. Subsequently, it was developed by many

authors and used in various fields. To use this concept in

Topology and Analysis, several researchers have defined

Fuzzy metric space in various ways. In this paper we deal

with the Fuzzy metric space defined by Kramosil and

Michalek [11] and modified by George and Veeramani

[20]. Recently, Grebiec [1] has proved fixed point results

for Fuzzy metric space. In the sequel, Singh and Chauhan

[12] introduced the concept of compatible mappings of

Fuzzy metric space and proved the common fixed point

theorem. Jungck et. al. [2] introduced the concept of

compatible maps of type (A) in metric space and proved

fixed point theorems. Using the concept of compatible

maps of type (A), Jain et. al. [18] proved a fixed point

theorem for six self maps in a fuzzy metric space. Singh et.

al. [7,8] proved fixed point theorems in a fuzzy metric

space. Recently in 2012, Jain et. al. [4,5] and Sharma et. al.

[6] proved various fixed point theorems using the concepts

of semi-compatible mappings, property (E.A.) and

absorbing mappings. The concept of occasionally weakly

compatible mappings in metric spaces is introduced by Al-

Thagafi and Shahzad [14] which is most general among

all the commutativity concepts. Recently, Khan and

Sumitra [15] extended the notion of occasionally weakly

compatible maps to fuzzy metric space.

In this paper, a fixed point theorem for six self maps

has been established using the concept of compatible maps

of type (A) occasionally weakly compatible mappings,

which generalizes the result of Cho [16].

For the sake of completeness, we recall some

definitions and known results in Fuzzy metric space.

2. Definitions, lemmas, Remarks,

Propositions

Definition 2.1. [10] A binary operation :

0,1 0,1 0,1 is called a -norm if is an

abelian topological monoid with unit such that

a b c d . Whenever a c and b d for .

[0,1].

Examples of -norms are:

Definition 2.2. [10] The -tuple is said to be a

Fuzzy metric space if X is an arbitrary set, is a

continuous t-norm and M is a Fuzzy set in 2 0,x

satisfying the following conditions:

For all and

(FM-1) (FM-2) (FM-3) (FM-4) (FM-5)

(FM-6)

Note that can be considered as the degree of

nearness between x and y with respect to t. We identify

with for all The following

example shows that every metric space induces a Fuzzy

metric space.

Example 2.1. [10] Let be a metric space. Define

and

, ,,

tM x y t

t d x y

for all

, and 0x y X t . Then is a Fuzzy metric space.

It is called the Fuzzy metric space induced by .


Definition 2.3. [10] A sequence nx in a Fuzzy metric

space is said to be a Cauchy sequence if and only

if for each , there exists 0n N such that

for all .

The sequence is said to converge to a point in

if and only if for each there exists

such that for all .

A Fuzzy metric space is said to be complete if

every Cauchy sequence in it converges to a point in it.

Definition 2.4. [12] Self mappings and of a Fuzzy

metric space are said to be compatible if and only

if for all , whenever

is a sequence in such that for some in

as .

Definition 2.5. [18] Self maps and of a Fuzzy metric

space are said to be compatible maps of type

if 1and 1 for

all , whenever is a sequence in such that

for some in as Definition 2.6. [15] Two maps and from a Fuzzy

metric space into itself are said to be

Occasionally weakly compatible (owc) if and only if there

is a point , which is coincidence point of and at

which and commute.

Remark 2.1. [18] The concept of compatible maps of type

) and occasionally weakly compatibility is more general

than the concept of compatible maps in a Fuzzy metric

space.

Proposition 2.1. [18] In a Fuzzy metric space

limit of a sequence is unique.

Lemma 2.1. [1] Let be a fuzzy metric space.

Then for all , is a non-decreasing

function.

Lemma 2.2. [16] Let be a fuzzy metric space. If

there exists such that for all

Lemma 2.3. [18] Let be a sequence in a fuzzy metric

space . If there exists a number such

that and

. Then is a Cauchy sequence in .

Proposition 2.2. [18] Let and be concept of

compatible maps of type ) of a complete fuzzy metric

space with continuous t-norm defined by

for all and

for some in . Then Lemma 2.4. [3] The only -norm satisfying

for all is the minimum -norm, that is for all

3. Main Result

Theorem 3.1. Let be a complete fuzzy metric

space and let and be mappings from into

itself such that the following conditions are satisfied:

(a) (b) (c) either or is continuous;

(d) is compatible maps of type and is

occasionally weakly compatible ;

(e) there exists such that for every

and

( , , ) ( , ,* ( , , )

* ( , , * ( , , ).

M Px Qy qt M ABx STy M Px ABx t v

M Qy STy t M Px STy t

Then and have a unique common fixed

point in .

Proof: Let ∈ X. From (a) there exist such

that and .

Inductively, we can construct sequences and in such that and

Step 1. Put and in (e), we get

2 2 1

2 2 1 2 2

2 1 2 1 2 2 1

2 2 1 2 1 2

2 2 2 1 2 1 2 1

2 2 1 2 1 2 2

, ,

, , * , ,

* , , * ( , , ).

, , * , ,

* , , * , ,

( , , )* ( , , ).

n n

n n n n

n n n n

n n n n

n n n n

n n n n

M Px Qx qt

M ABx STx t M Px ABx t

M Qx STx t M Px STx t

M y y t M y y t

M y y t M y y t

M y y t M y y t

From lemma 2.1 and 2.2, we have

Similarly, we have

Thus, we have

1 1

2 1

1 2

( , , ) ( , , / )

( , , / 2)

... ... ... ...

( , , / ) 1 ,

n n n n

n n

n

M y y t M y y t q

M y y t q

M y y t q asn

and hence as for any .

For each and , we can choose

such that

for all .

For , we suppose . Then we have

1

1 2

1

( , , ) ( , , / )

* ( , , /

* * ( , , /

1 * 1 * * 1

( , , ) (1 )

n m n n

n n

m m

n m

M y y t M y y t m n

M y y t m n

M y y t m n

m n times

M y y t

and hence is a Cauchy sequence in .

Since is complete, converges to some

point . Also its subsequence’s converges to the

same point i.e.

i.e., and (1)

and (2)


Case I. Suppose is continuous.

Since is continuous, we have

and

As is compatible pair of type , we have

Step 2. Put and in (e), we get

2 2 1

2 2 1 2 2

2 1 2 1 2 2 1

, ,

( , , )* ( , , )

* , , * ( , , ).

n n

n n n n

n n n n

M PABx Qx qt

M ABABx STx t M PABx ABABx t

M Qx STx t M PABx STx t

Taking , we get

( , , ) ( , , )* ( , ,

* ( , , )* ( , , )

( , , )* ( , , )

M ABz z qt M ABz z t M ABz ABz

M z z t M ABz z t

M ABz z t M ABz z t

Therefore, by using lemma 2.2, we get

(3)

Step 3. Put and in (e), we have

2 1

2 1

2 1 2 1 2 1

( , ,

( , , )* ( , , )

* , , * ( , , ).

n

n

n n n

M Pz Qx qt

M ABz STx t M Pz ABz t

M Qx STx t M Pz STx t

Taking and using equation (1), we get

( , , ) ( , , )* ( , , )

* ( , , )* ( , , )

( , , )* ( , , )

M Pz z qt M z z t M Pz z t

M z z t M Pz z t

M Pz z t M Pz z t

i.e. Therefore, by using lemma 2.2, we get

Therefore,

Step 4. Putting and in condition (e),

we get

2 1

2 1

2 1 2 1 2 1

( , , )

( , , )* ( , , )

* , , * ( , , ).

n

n

n n n

M PBz Qx qt

M ABBz STx t M PBz ABBz t

M Qx STx t M PBz STx t

As , so we have

( )( )

( )( ) ( ) .

P Bz B Pz Bzand AB Bz

BA Bz B ABz Bz

Taking and using (1), we get

( , , )

( , , )* ( , , )* ( , , )* ( , , )

( , , )* ( , , )

M Bz z qt

M Bz z t M Bz Bz t M z z t M Bz z t

M Bz z t M Bz z t


and also we have

Therefore,

Az Bz Pz z (4)

Step 5. As , there exists such that

.

Putting and in (e), we get

2

2 2 2

( , , )

( , , )* ( , , )

* , , * ( 2 , , ).

n

n n n

M Px Qu qt

M ABx STu t M Px ABx t

M Qu STu t M Px n STu t

Taking and using (1) and (2), we get

( , , ) ( , , )* ( , , )

* ( , , )* ( , , )* ( , , )

( , , )

M z Qu qt M z z t M z z t

M Qu z t M z z t M z z t

M Qu z t

i.e. Therefore, by using lemma 2.2, we get Hence

Since is occasionally weakly

compatible, therefore, by proposition (2.2), we have

Thus

Step 6. Putting and in (e), we get

2

2 2 2

2

, ,

, , * ( , , )

* , , * ( , , ).

n

n n n

n

M Px Qz qt

M ABx STz t M Px ABx t

M Qz STz t M Px STz t

Taking and using (2) and step 5, we get

( , , ) ( , , )* ( , , )

* ( , , )* ( , , )

( , , )* ( , , )

M z Qz qt M z Qz t M z z t

M Qz Qz t M z Qz t

M z Qz t M z Qz t


Qz = z.

Step 7. Putting and in (e), we get

2

2 2 2

2

( , , )

( , , )* ( , , )

* , , * ( , , ).

n

n n n

n

M Px QTz qt

M ABx STTz t M Px ABx t

M QTz STTz t M Px STTz t

As and , we have

and ( ) ( ) .

QTz TQz Tz

ST Tz T STz TQz Tz

Taking , we get

( , , ) ( , , )* ( , , )

* ( , , )* ( , , )

( , , )* ( , , )

M z Tz qt M z Tz t M z z t

M Tz Tz t M z Tz t

M z Tz t M z Tz t

i.e.


Now implies

Hence

.Sz Tz Qz z (5)

Combining (4) and (5), we get

.Az Bz Pz Qz Tz Sz z

Hence, is the common fixed point of and

.


Case II. Suppose is continuous.

As is continuous,

and

As is compatible pair of type ,

Step 8. Putting and in condition

(e), we have

2 2 1

2 2 1 2 2

2 1 2 1 2 2 1

( , , )

( , , )* ( , , )

* , , * ( , , ).

n n

n n n n

n n n n

M PPx Qx qt

M ABPx STx t M PPx ABPx t

M Qx STx t M PPx STx t

Taking , we get

( , , ) ( , , )* ( , , )

* ( , , )* ( , , )

( , , )* ( , , )

M Pz z qt M Pz z t M Pz Pz t

M z z t M Pz z t

M Pz z t M Pz z t

i.e. Therefore by using lemma 2.2, we have

Further, using steps 5,6,7, we get

Step 9. As , there exists such that

Put and in (e), we get

2 1

2 1

2 1 2 1 2 1

( , , )

( , , )* ( , , )

* , , * ( , , ).

n

n

n n n

M Pw Qx qt

M ABw STx t M Pw ABw t

M Qx STx t M Pw STx t

Taking , we get

( , , ) ( , , )* ( , , )

* ( , , )* ( , , )

( , , )* ( , , )

M Pw z qt M z z t M Pw z t

M z z t M Pw z t

M Pw z t M Pw z t


Therefore,

As is compatible pair of type , then by

proposition (2.2), we have

Also, from step 4, we get .

Further, using steps 5, 6, 7, we get

i.e. is the common fixed point of the six maps

and in this case also.

Uniqueness: Let be another common fixed point of

and .

Then .

Put and in (e), we get

, , , , * , ,

* ( , , )* ( , , ).

M Pz Qu qt M ABz STu t M Pz ABz t

M Qu STu t M Pz STu t

Taking , we get

( , , ) ( , , )* ( , , )

* ( , , )* ( , , )

( , , )* ( , , )

M z u qt M z u t M z z t

M u u t M z u t

M z u t M z u t

i.e. Therefore by using lemma (2.2), we get .

Therefore is the unique common fixed point of self

maps and .

Remark 3.1. If we take , the identity map on

in theorem 3.1, then condition is satisfied trivially

and we get

Corollary 3.1. Let be a complete fuzzy metric

space and let and be mappings from into itself

such that the following conditions are satisfied:

(a) ;

(b) either or is continuous;

(c) is compatible maps of type and is

occasionally weakly compatible;

(d) there exists such that for every

and

( , , ) ( , , )* ( , , )

* ( , , )* ( , , ).

M Px Qy qt M Ax Sy t M Px Ax t

M Qy Sy t M Px Sy t

Then and have a unique common fixed point

in .

Remark 3.2. In view of remark 3.1, corollary 3.1 is a

generalization of the result of Cho [16] in the sense that

condition of compatibility of the pairs of self maps has

been restricted to compatibility of type occasionally

weakly compatible and only one map of the first pair is

needed to be continuous.

Acknowledgement

Authors are thankful to the referee for his valuable

comments.

References

[1] George and P. Veeramani, On some results in Fuzzy metric spaces,

Fuzzy Sets and Systems 64 (1994), 395-399.

[2] Jain and B. Singh, A fixed point theorem for compatible mappings

of type (A) in fuzzy metric space, Acta Ciencia Indica, Vol.

XXXIII M, No. 2 (2007), 339-346.

[3] Jain, M. Sharma and B. Singh, Fixed point theorem using

compatibility of type (β) in Fuzzy metric space, Chh. J. Sci. &

Tech., Vol. 3 & 4, (2006- 2007), 53-62.

[4] Jain, V.H. Badshah, S.K. Prasad, Fixed Point Theorem in Fuzzy

Metric Space for Semi-Compatible Mappings, Int. J. Res. Rev.

Appl. Sci. 12 (2012), 523-526.

[5] Jain, V.H. Badshah, S.K. Prasad, The Property (E.A.) and The

Fixed Point Theorem in Fuzzy Metric, Int. J. Res. Rev. Appl. Sci.

12 (2012), 527-530.

[6] Sharma, A. Jain, S. Chaudhary, A note on absorbing mappings and

fixed point theorems in fuzzy metric space, Int. J. Theoretical

Appl. Sci. 4 (2012), 52-57.

[7] Singh, A. Jain, A.K. Govery, Compatibility of type and fixed

point theorem in Fuzzy metric space, Appl. Math. Sci. 5 (2011),

517-528.

[8] Singh, A. Jain, A.K. Govery, Compatibility of type (A) and fixed

point theorem in Fuzzy metric space,Int. J. Contemp. Math. Sci. 6

(2011), 1007-1018.

[9] Singh and M.S. Chouhan, Common fixed points of compatible

maps in Fuzzy metric spaces, Fuzzy sets and systems, 115 (2000),

471-475.


[10] E.P. Klement, R. Mesiar and E. Pap, Triangular Norms, Kluwer

Academic Publishers.

[11] G. Jungck, P.P. Murthy and Y.J. Cho, Compatible mappings of

type (A) and common fixed points, Math. Japonica, 38 (1993),

381-390.

[12] I.Kramosil and J. Michalek, Fuzzy metric and statistical metric

spaces, Kybernetica 11(1975), 336-344.

[13] L. A. Zadeh, Fuzzy sets, Inform and control 89 (1965), 338-353.

[14] M.A. Al-Thagafi, N.A. Shahzad, A note on occasionally weakly

compatible maps, Int. J. Math. anal. 3(2009), 55-58.

[15] M.A. Khan, Sumitra, Common fixed point theorems for

occasionally weakly compatible maps in fuzzy metric spaces, Far

East J. Math. Sci., 9 (2008), 285-293.

[16] S.H., Cho, On common fixed point theorems in fuzzy metric spaces,

J. Appl. Math. & Computing Vol. 20 (2006), No. 1-2, 523-533.

[17] S.N. Mishra, N. Mishra and S.L. Singh, Common fixed point of

maps in fuzzy metric space, Int. J. Math. Math. Sci. 17(1994), 253-

258.

[18] M. Grebiec, Fixed points in Fuzzy metric space, Fuzzy sets and

systems, 27(1998), 385-389.

[19] Y.J. Cho, Fixed point in Fuzzy metric space, J. Fuzzy Math.

5(1997), 949-962.

[20] Y.J. Cho, H.K. Pathak, S.M. Kang and J.S. Jung, Common fixed

points of compatible mappings of type (β) on fuzzy metric spaces,

Fuzzy sets and systems, 93 (1998), 99-111.




DOI:10.12691/tjant-3-3-3

Coefficient Estimates for Starlike and Convex Classes of

-fold Symmetric Bi-univalent Functions

S. Sivasubramanian1, P.Gurusamy

2,*

1Department of Mathematics, University College of Engineering Tindivanam, Anna University, Tindivanam, India 2Department of Mathematics, Velammal Engineering College, Surapet, Chennai, India



Abstract In an article of Pommerenke [10] he remarked that, for an -fold symmetric functions in the class ,

the well known lemma stated by Caratheodary for a one fold symmetric functions in still holds good. Exploiting

this concept, we introduce certain new subclasses of the bi-univalent function class in which both and are

-fold symmetric analytic with their derivatives in the class of analytic functions. Furthermore, for functions in

each of the subclasses introduced in this paper, we obtain the coefficient bounds for and We remark

here that the concept of -fold symmetric bi-univalent is not in the literature and the authors hope it will make the

researchers interested in these type of investigations in the forseeable future. By the working procedure and the

difficulty involved in these procedures, one can clearly conclude that there lies an unpredictability in finding the

coefficients of a -fold symmetric bi-univalent functions.

Keywords: analytic functions, univalent functions, bi-univalent functions, -fold symmetric functions,

subordination

Cite This Article: S. Sivasubramanian, and P.Gurusamy, “Coefficient Estimates for Starlike and Convex

Classes of -fold Symmetric Bi-univalent Functions.” Turkish Journal of Analysis and Number Theory, vol. 3,

no. 3 (2015): 83-86. doi: 10.12691/tjant-3-3-3.

1. Introduction

Let denote the class of functions of the form:

∑

(1)

which are analytic in the open unit disk Further, by we mean the class of all functions in

which are univalent in . For more details on univalent

functions, see [3]. It is well known that every function

has an inverse , defined by

(2)

and

(

) (3)

Indeed, the inverse function may have an analytic

continuation to , with

1 2 2 3

2 2 3

3 42 2 3 4

( ) (2 )

(5 5 ) .

f w w a w a a w

a a a a w

(4)

A function is said to be bi-univalent in if both

and are univalent in . Let denote the class of bi-

univalent functions in , given by equation (1). An

analytic function is subordinate to an analytic function ,

written , provided there is an analytic

function defined on with and

satisfying . Lewin [8] investigated the

class of bi-univalent functions and obtained a bound

. Motivated by the work of Lewin [8],

Brannan and Clunie [1] conjectured that √ . Some

examples of bi-univalent functions are

(

) and

(see also the work of Srivastava et al. [11]).

The coefficient estimate problem for each of the following

Taylor-Maclaurin coefficients: is

still open([11]). In recent times, the study of bi-univalent

functions gained momentum mainly due to the work of

Srivastava et al. [11]. Motivated by this, many researchers

(see [4,11,12,13,14,15,17]) recently investigated several

interesting subclasses of the class and found non-sharp

estimates on the first two Taylor-Maclaurin coefficients.

For each function in , the function √ is

univalent and maps the unit disk into a region with -

fold symmetry. A function is -fold symmetric (see [10])

if it has the normalized form

∑

(5)

and we denote the class of -fold symmetric univalent

functions by , which are normalized by the above series

expansion. In fact, the functions in the class are one fold

symmetric. Analogous to the concept of -fold symmetric

univalent functions, one can think of the concept of -

fold symmetric bi-univalent functions in a natural way.

Each function in the class in , generates an -fold

symmetric bi-univalent function for each integer . The

normalized form of is given as in (5) and is given as

follows.


1 2 2 11 1 2 1

31 3 1

1 2 1 3 1

( ) ( 1)

1( 1)(3 2)

.2

(3 2)

m mm m m

m m

m m m

g w w a w m a a w

m m aw

m a a a

(6)

where . We denote the class of -fold symmetric

bi-univalent functions by . For , the formula (6)

coincides with the formula (4) of the class Denote also,

by , the class of analytic functions of the form

such that ( ) in . In

view of Pommerenke [10], the -fold symmetric function

in the class is of the form

(7)

It is assumed that is an analytic functions with

positive real part in the unit disk , with and is symmetric with respect to the

real axis. Such a function has a series expansion of the

form

(8)

Suppose that and are analytic functions in

the unit disk with and , and suppose that

(9)

and

(10)

We observe that

2 2

2 21, 1 , 1and 1 .m m m m m mb b b c c c (11)

By a simple computations, we have

2 21 1 2 21 ( ) ( 1)

m mm m mu z B b z B b B b z z (12)

and

2 21 1 2 21 ( ) ( 1).

m mm m mv w B c w B c B c w w (13)

Motivated essentially by the work of Ma and Minda [9],

we introduce some new subclasses of -fold symmetric

bi-univalent functions and obtain coefficient bounds of

and for functions in these classes. The

results presented in this paper improve the earlier results

of Frasin and Aouf [4], Srivastava et al. [11] for the case

of one fold symmetric functions.

2. Coefficient Estimates for the Function

Class

Definition 2.1 A function , given by (5), is said to

be in the class , if the following conditions are

satisfied:

1

',

'and ,

m

zf zf z

f z

wg ww g w f w

g w

where the function is defined by (6).

For the special choices of the function and for the

choice of our class reduces to the following.

1. For and (

)

,

( (

)

) the class of

strongly bi-starlike functions of order studied by

Brannan and Taha [2].

2. For and

,

( (

))

the class of

bi-starlike functions of order studied by Brannan and

Taha [2].

We first state and prove the following theorem.

Theorem 2.1 Let given by (5), be in the class

. Then

√

√ | |

(14)

and

12 12

22 11 2 1 2 1

2 12 2

1 2 1

1

2

1

2

m

m Bif B B

m

a m B B B B Bif B B

m B B B

(15)

Proof. Let and . Then there are

analytic functions , with

satisfying

( )

( ) (16)

Since

and

2

211

2 1

' 2 11 ,

2

m mmm

m

wg w m m ama w w

g w ma

it follows from (12), (13) and (16) that

(17)

(18)

(19)

and

(20)

From (17) and (19), we get

(21)

By adding (18) and (20) and in view of the

computations using (17) and (21), we get

(22)

Further, (21), (22), together with (11), gives

(23)

Now from (17) and (23), we get


√

√ | |

as asserted in (14).

By simple calculations from (18) and (20) using with

the equations (17) and (21), we get

2 22 1 1 2 1 2 24 1 2 2 1m m m mm a m B b B c m B b (24)

Then using the equation (11) in (24), we get

(25)

Since

| |

(26)

substituting (26) in (25), we get

12 12

22 11 2 1 2 1

2 12 2

1 2 1

1

2

1

2

m

m Bif B B

m

a m B B B B Bif B B

m B B B

as asserted in (15). This completes the proof of Theorem

2.1.

For the case of one fold symmetric functions, Theorem

2.1 reduces to the coefficient estimates for Ma-Minda bi-

starlike functions in Srivastava et al [11].

Corollary 2.1 Let , given by (5), be in the class

. Then

√

√| |

(27)

and

(28)

For the case of one fold symmetric functions and for the

class of strongly starlike functions , the function is

given by

2 211 2 2 , 0 1

1

zz z z

z

(29)

which gives and . Hence Theorem 2.1

reduce to the result in Brannan and Taha [2].

Corollary 2.2 [2] Let , given by (5), be in the class

. Then

√ (30)

and

(31)


class of strongly starlike functions , the function is

given by

then , and the Theorem 2.1 reduce to

the result in Brannan and Taha [2].


. Then

√ (32)

and

√ (33)

3. Coefficient Bound for the Function

Class

Definition 3.1 A function , given by (5), is said to

be in the class , if the following conditions are

satisfied:

and

where the function is defined by (6).

For one fold symmetric, a function in the class

is called bi-Mocanu-convex function of Ma-

Minda type. For the special choices of the function and for the choice of our class reduces to the

following.

1. For and

( (

)

) the class of strongly bi-

convex functions of order studied by Brannan and Taha

[2].

2. For and

( (

)) the class of

bi-convex functions of order studied by Brannan and

Taha [2].

Theorem 3.1 Let given by (5), be in the class

Then

√

√ | |

(34)

and

12 12

21 2 1

2 12 1

2 121 22

1

2

1

1

12

1

m

Bif B B

m

B m B B

am B B

if B BB m B

m

m B

(35)

Proof. Let . Then there are analytic

functions , with satisfying

( )

( ) (36)

Since

1

2 2 22 1 1

''1 1 1

'

2 1 2 1

mm

mm m

zf zm m a z

f z

m m a m m a z

and


1

21 2

2 1

''1 1 1

'

1 3 1

2 1 2

mm

m m

m

wg wm m a w

g w

m m m aw

m m a

from (12), (13), and (36), we get

(37)

2 2 2

2 1 1 1 2 22 1 2 1 ,m m m mm m a m m a B b B b (38)

(39)

and

2

1 2 1

21 2 2

1 3 1 2 1 2

.

m m

m m

m m m a m m a

B c B c

(40)

From (37) and (39), we get

(41)

By adding the equations (38) and (40), in view of

computations using (37) and (41), we get

2 2 21 2 1

31 2 2

2 1 1

.

m

m m

m m B m B a

B b c

(42)

Further, from the equations (41), (42), together with

(11), we have

22 2 2 31 2 1 11 1 1 .m mm m B m B a B b (43)

Now from (37) and (43), we get

√

√ | |

as asserted in(34). By simple calculations from (38) and

(40) using with the equations (37) and (41), we get

22 1

21 2 1 2 2

4 1 2

1 3 1 2 1 2 .

m

m m m

m m a

m B b m B c m B b

(44)

Then using the equation (11) in (44), we get

(45)

Since

| |

(46)

substituting (46) in (45), we get

12 12

22 1 1 2 1 2 1

2 12 2

1 2 1

2

1 1

2 1 1

m

Bif B B

m

a B m B B m B Bif B B

m B m B m B

as asserted in (35).

For one fold symmetric functions then, Theorem 3.1

gives the coefficient for Ma-Minda bi-convex functions in

Brannan and Taha [2]


. Then

√

√| |

and


class of strongly starlike functions, the function is given

by

then , and the Theorem 3.1 reduce to

the result in Brannan and Taha[2].


. Then

and

References

[1] D. A. Brannan and J. G. Clunie, Aspects of contemporary complex

analysis, Academic Press, London, 1980.

[2] D. A. Brannan and T. S. Taha, On some classes of bi-univalent

functions, Stud.Univ.Babes-Bolyai Math. 31(1986), 70-77.

[3] P. L. Duren, Univalent functions, Springer-Verlag, New York,

Berlin, Hiedelberg and Tokyo, 1983.

[4] B. A. Frasin and M. K. Aouf, New subclasses of bi-univalent

functions, Appl. Math. Lett. 24 (2011), 1569-1573.

[5] J. Sokol, On a condition for - starlikeness, J. Math. Anal. Appl.

352(2009), 696-701.

[6] M . K . Aouf , J . Sokol and J . Dziok, On a subclass of strongly

starlike functions, Appl. Math. Lett. 24 (2011), 27-32.

[7] J . Sokol, A certain class of starlike functions, Comput. Math.

Appl, 62(2011), 611-619.

[8] M. Lewin, On a coefficient problem for bi-univalent functions,

Proc. Amer. Math. Soc. 18 (1967), 63-68.

[9] W. C. Ma, and D. Minda, A unified treatment of some special

classes of unvalent functions, in Proceedings of the Conference on

complex analysis. Tianjin (1992), 157-169.

[10] Ch.Pommerenke, On the coefficients of close-to-convex functions,

Michigan. Math. J. 9 (1962), 259-269.

[11] H. M. Srivastava, A. K. Mishra and P. Gochhayat, Certain

subclasses of analytic and bi-univalent functions, Appl. Math. Lett.

23 (2010), 1188-1192.

[12] H. M. Srivastava, S. Bulut, M. Cagler and N. Yagmur, Coefficient

estimates for a general subclass of analytic and bi-univalent

functions, Filomat. 27 (2013), 831-842.

[13] Q.-H. Xu, H. M. Srivastava and Z. Li, A certain subclass of

analytic and close-to-convex functions, Appl. Math. Lett. 24

(2011), 396-â€“401.

[14] Q.-H. Xu, Y.-C. Gui and H. M. Srivastava, Coefficient estimates

for a certain subclass of analytic and bi-univalent functions, Appl.

Math. Lett. 25 (2012), 990-994.

[15] Q.-H. Xu, H.-G. Xiao and H. M. Srivastava, A certain general

subclass of analytic and bi-univalent functions and associated

coefficient estimate problems, Appl. Math. Comput. 218 (2012),

11461-11465.

[16] Zhigang Peng and Qiuqiu Han, On the Coefficients of several

classes of bi-uivalent functions, Acta. Math. Sci. 34B(1) (2014),

228-240.

[17] H. Tang, G-T. Deng and S-H. Li, Coefïcient estimates for new

subclasses of Ma-Minda bi-univalent functions, J. Inequal. Appl.

2013, 2013:317, 1-10.




DOI:10.12691/tjant-3-3-4

Schur-geometric and Schur-harmonic Convexity of an

Integral Mean for Convex Functions

Jian Sun1, Zhi-Ling Sun

1, Bo-Yan Xi

1, Feng Qi

2,3,*

1College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China 2Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin City, China;

3Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, China


Received April 11, 2015; Accepted June 20, 2015

Abstract In the paper, Schur-geometric and Schur-harmonic convexity of an integral mean for convex functions

are established.

Keywords: Schur-convex function; Schur-geometrically convex function; Schur-harmonically convex function;

inequality; generalized logarithmic mean

Cite This Article: Jian Sun, Zhi-Ling Sun, Bo-Yan Xi, and Feng Qi, “Schur-geometric and Schur-harmonic

Convexity of an Integral Mean for Convex Functions.” Turkish Journal of Analysis and Number Theory, vol. 3,

no. 3 (2015): 87-89. doi: 10.12691/tjant-3-3-4.

1. Introduction

In [3], N. Elezović and J. Pečarić established the

following theorem.

Theorem A ([3]). Let :f I R R and ,a b I . Then

1( )d , ,

( , )

( ),

b

af x x a b

F a b b a

f a a b

is Schur-convex (Schur-concave) on 2I if and only if f

is convex (concave) on .I

In [7,10], Theorem A was generalized as the following

theorem.

Theorem B ([7,10]). Let be a continuous function

and a positive continuous weight on an interval . Then

the weighted arithmetic mean of with weight defined

by

( ) ( )d, ,

( , )( )d

( ),

y

x

y

x

p t f t tx y

G x yp t t

f x x y

is Schur-convex (Schur-concave) on if and only if

( ) ( )d ( ) ( ) ( ) ( )

( )+ ( )( )d

y

x

y

x

p t f t t p x f x p y f y

p x p yp t t

holds (reverses) for all .

For more information on this topic, please refer to

[5,8,9] and closely-related references therein.

In this paper, we discuss Schur-geometric and Schur-

harmonic convexity of the mean and obtain two

results which generate Theorem A.

2. Definitions and Lemmas

In order to prove our main results we need the

following definitions and lemmas.

Definition 1 ([4]). Let I R and 1( , , )nx x x ,

1( , , ) ,nny y y I and let : .nI R

(1) x is said to be majorized by y (in symbols x y )

if 1 1

k ki ii i

x y

for 1,2, , 1k n and

[ ]1 1

n nii i

x y

[i] , where [1] [ ]nx x and

[ ]ny y [1] are rearrangements of x and y in a

descending order.

(2) x y means i ix y for all 1,2, , .i n is said

to be increasing if x y implies ( ) ( )x y . is said

to be decreasing if and only is increasing.

(3) is said to be a Schur-convex function on nI if

x y on nI implies ( ) ( )x y . is said to be a

Schur-concave function on nI if and only is Schur-

convex function.

Definition 2 ([1,2]). Let 1( , , )nx x x , 1( , , )ny y y

+n nI R and : nI R and let 1ln (ln , , ln ),nx x x

1 1 1

1, ,

x x xn .

(1) is said to be a Schur-geometrically convex

function on nI if ln lnx y on nI implies ( ) ( )x y .


is said to be a Schur-geometrically concave function on

nI if and only is Schur-geometrically convex

function.

(2) is said to be a Schur-harmonically convex

function on nI if 1 1x y

on nI implies ( ) ( )x y .

is said to be a Schur-harmonically concave function on nI

if and only is Schur-harmonically convex function.

Lemma 2.1([1]). Let 2 2+: I R R be a continuous

function on 2I and differentiable in interior of 2I . Then

is Schur-geometrically convex (Schur-geometrically

concave) if and only if it is symmetric and

( )0b a b ab a

for all , .a b I

Lemma 2.2 ([2]). Let 2 2+: I R R be a

continuous function on 2I and differentiable in interior of 2I . Then is Schur-harmonically convex (Schur-

harmonically concave) if and only if it is symmetric and

2 2 ( )0b a b ab a

for all , .a b I

For two positive numbers 0a and 0b , define

( , ) ,2

( , ) ,

2( , )

a bA a b

G a b ab

abH a b

a b

and

11 1

, , 1,( , ) ( 1)( )

, .

rr r

r

b aa b r

L a b r b a

a a b

It is well known that ( , )A a b , ( , )G a b , ( , )H a b and

( , )sL a b are respectively called the arithmetic, geometric,

harmonic and generalized logarithmic means of a and .b

Lemma 2.3 ([6]) ( , )rL a b is increasing function on

2+( , )a b R .

In this paper, we will prove that the function ( , )F a b is

Schur-geometrically convex and Schur-harmonically

convex on 2+R .

3. Main Results

Theorem 3.1. Let :f I R R and F be defined in

Theorem A.

(i). If f is convex and increasing on I , then F is

Schur-geometrically convex on 2I .

(ii). If f is concave and decreasing on I , then F is

Schur-geometrically concave on 2I .

Proof. If ,a b I and a b , we have ( , ) ( ).F a a f a

For all , ,a b I a b , a straightforward computation

gives

1 1, ,

1 1, .

Ff b F a b

b b a b a

Ff a F a b

a b a b a

(3)

If f is convex and increasing on I , by the inequality

(2), we obtain

( ) ( ) ( ) ( , )

( ) ( ) ( )d

12 ( ) 2 ( ) ( )( ( ) ( ))

2

1( )( ( ) ( )) 0.

2

b

a

F Fb a b a

b a

af a bf b a b F a b

a baf a bf b f x x

b a

af a bf b a b f a f b

b a f b f a

(4)

Hence, ( , )F a b is Schur-geometrically convex on 2I . If

f is concave and decreasing on I , then the inequality (4)

is reversed. According to Lemma 2.1, it follows that

( , )F a b is Schur-geometrically concave 2I . This

completes the proof of Theorem 3.1.

Theorem 3.2. Let :f I R R and F be defined in

Theorem A.

(i). If f is convex and increasing on I , then F is

Schur-harmonically convex on 2I .

(ii). If f is concave and decreasing on I , then F is

Schur-harmonically concave on 2I .

Proof . If ,a b I and a b , we have ( , ) ( ).F a a f a

For all , ,a b I a b , if f is convex and increasing,

using inequality (3) and (2), we get

2 2

2 22 2

2 22 2

2 2

( )

( ) ( ) ( )d

( ) ( ) ( ) ( )2

1= ( ) ( ) ( ) 0.

2

b

a

F Fb a b a

b a

a ba f a b f b f x x

b a

a ba f a b f b f a f b

b a f b f a

(5)

Therefore, ( , )F a b is Schur-harmonically convex function

on 2I . If f is concave and decreasing function on I ,

then the inequality (5) is reversed. According to Lemma

2.2, it follows that ( , )F a b is Schur-harmonically concave

function on 2I . The proof of Theorem 3.2 is complete.


4. Applications

Theorem 4.1. For 0a and 0b , if 1r , then

( , )rL a b is Schur-geometrically convex and Schur-

harmonically convex.

Proof. Taking ( ) rf x x for all x R , if a b , it

follows that

+1 +11 1( , ) ( )d d

( 1)( )

r rb b r

a a

b aF a b f x x x x

b a b a r b a

and ( ) rf x x is convex increasing on R for 1r .

Therefore, by Theorem 3.1 and 3.2, we have

+1 +1

, ,( 1)( )( , )

,

r r

r

b aa b

r b aF a b

a a b

is Schur-geometrically convex and Schur-harmonically

convex on 2R for 1r , then ( , )rL a b is Schur-

geometrically convex and Schur-harmonically convex on 2R for 1r . Thus, Theorem 4.1 is proved.

Corollary 4.1.1. For 0b a and 1r , define

(1 )au ta t b , (1 )av t a tb , 1t tgu a b ,

1 t tgv a b , 1 1(1 )hu ta t b ， and

1 1(1 )hv t a tb for (0,1)t . Then

(1) when (0,1)t and 1/ 2t , we have

1/1/+1 +1( +1) ( +1)

1 1

1/+1 +1

( 1)( )( 1)( )

( , );( 1)( )

rrr rr rg gh h

g gh h

rr ra a

r

a a

u vu v

r u vr u v

u vL a b

r u v

(2) when =1/ 2t , we have

( , ) ( , ) ( , )= ( , ).rH a b G a b A a b L a b

Proof. When =1/ 2t , it is easy to obtain that

( , ) ( , ) ( , ).rH a b G a b L a b When (0,1)t and 1/ 2t ,

by Corollary 2 in [6] and Lemma 2.3, Corollary 4.1.1 is

thus proved.

Acknowledgements

The authors thank the anonymous referees for their

careful corrections to and valuable comments on the

original version of this paper.

Support

This work was partially supported by the National

Natural Science Foundation of China under Grant No.

11361038 and by the Inner Mongolia Autonomous Region

Natural Science Foundation Project under Grant No.

2015MS0123 and No. 2014BS0106, China.

References

[1] Y.-M. Chu, X.-M. Zhang, and G.-D. Wang, The Schur

geometrical convexity of the extended mean values, J. Convex

Anal. 15 (2008), no. 4, 707-718.

[2] W.-F. Xia and Y.-M. Chu, Schur-convexity for a class of

symmetric functions and its applications, J. Inequal. Appl. 2009

(2009), Article ID 493759, 15 pages.

[3] N. Elezović and J. Pečarić, A note on Schur-convex functions,

Rocky Mountain J. Math. 30 (2000), no. 3, 853-856.

[4] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization

and its Application, New York: Academies Press, 2011.

[5] C. Mortici, Arithmetic mean of values and value at mean of

arguments for convex functions, ANZIAM J. 50 (2008), no. 1,

137-141.

[6] F. Qi and Q.-M. Luo, A simple proof of monotonicity for extended

mean values, J. Math. Anal. Appl. 224 (1998), 356-359.

[7] F. Qi, J. Sándor, S. S. Dragomir, and A. Sofo, Notes on the Schur-

convexity of the extended mean values, Taiwanese J. Math. 9

(2005), no. 3, 411-420.

[8] H.-N. Shi, Schur-convex functions related to Hadamard-type

inequalities, J. Math. Inequal. 1 (2007), no. 1 127-136.

[9] H.-N. Shi, D.-M. Li and C. Gu, The Schur-convexity of the mean

of a convex function, Appl. Math. Lett. 22 (2009), no. 6, 932-937.

[10] H.-N. Shi, S.-H. Wu, and F. Qi, An alternative note on the Schur-

convexity of the extended mean values, Math. Inequal. Appl. 9

(2006), no. 2, 219-224.




DOI:10.12691/tjant-3-3-5

Symmetric Identities Involving q-Frobenius-Euler

Polynomials under Sym (5)

Serkan Araci1,*

, Ugur Duran2, Mehmet Acikgoz

2

1Department of Economics, Faculty of Economics, Administrative and Social Sciences, Hasan Kalyoncu University, Gaziantep,

TURKEY 2University of Gaziantep, Faculty of Science and Arts, Department of Mathematics, Gaziantep, TURKEY


Received April 21, 2015; Accepted June 26, 2015

Abstract Following the definition of q-Frobenius-Euler polynomials introduced in [3], we derive some new

symmetric identities under sym (5), also termed symmetric group of degree five, which are derived from the

fermionic p-adic q-integral over the p-adic numbers field.

Keywords: Symmetric identities, q-Frobenius-Euler polynomials, Fermionic p-adic q-integral on p Invariant

under S5

Cite This Article: Serkan Araci, Ugur Duran, and Mehmet Acikgoz, “Symmetric Identities Involving q-

Frobenius-Euler Polynomials under Sym (5).” Turkish Journal of Analysis and Number Theory, vol. 3, no. 3

(2015): 90-93. doi: 10.12691/tjant-3-3-5.

1. Introduction

As it is known, the Frobenius-Euler polynomials

nH x for with 1 are defined by means of

the power series expansion at 0t

0

1.

!

nxt

n tn

tH x e

n e

(1.1)

Taking x = 0 in the Eq. (1.1), we have 0 :n nH H

that is widely known as n-th Frobenius-Euler number cf.

[3,4,5,8,17,18,21]. Let p be chosen as a fixed odd prime number.

Throughout this paper, we make use of the following

notations: p denotes topological closure of ,

denotes the field of rational numbers, p denotes

topological closure of , and p indicates the field of

p-adic completion of an algebraic closure of p . Let

be the set of natural numbers and * 0 .

For d an odd positive number with (p,d) = 1, let

1: lim / Nd p

nX X dp and X

and

| modN Npa dp x X x a dp

where a lies in 0 Na dp . See, for details,

[1,2,3,4,6-17].

The normalized absolute value according to the theory

of p-adic analysis is given by 1p

p p . q can be

considered as an indeterminate a complex number q

with 1q , or a p-adic number pq with

1

11 pp

q p

and exp logxq x q for 1.p

x It is

always clear in the content of the paper. Throughout this paper, we use the following notation:

1

.1

x

q

qx

q

(1.2)

which is called q-extension of x. It easily follows that

1limq qx x for any x.

Let f be uniformly differentiable function at a point

,pa which is denoted by .pf UD Then the

p-adic q-integral on p (or sometimes called q-

Volkenborn integral) of a function f is defined by Kim [10]

1

0

1lim .

Np

xq q Np N

xq

I f f x d x f x qp

(1.3)

It follows from the Eq. (1.3) that

1 11

1

0

lim

lim 1 .

qq p

Npx

Nx

I f I f f x d x

f x

(1.4)


Thus, by the Eq. (1.4), we have

1

1 11 1

0

1 2 1n

n n sn

s

I f I f f s

where , .nf x f x n n . For the applications of

fermionic p-adic integral over the p-adic numbers field,

see the references, e. g., [1,2,3,4,6,7,9,11,12,16]. In [3], the q-Frobenius-Euler polynomials are defined

by the following p-adic fermionic q-integral on p , with

respect to 1 :

1,

1

|

1.

2

n q

nyq

p

H x

x y d y

(1.5)

Upon setting x = 0 into the Eq. (1.5) gives

, ,0 :n q n qH H which are called n-th q-Frobenius-Euler

number.

By letting 1q in the Eq. (1.5), it yields to

1 1,

1

1

lim | : |

1.

2

n q nq

ny

p

H x H x

x y d y

Recently, many mathematicians have studied the

symmetric identities on some special polynomials, see, for

details, [1,6,7,9,12]. Some of mathematicians also

investigated some applications of Frobenius-Euler

numbers and polynomials (or its q-analog) cf.

[3,4,5,13,14,15,16]. Moreover, Frobenius-Euler numbers

at the value λ = 1 in (1.1) are Euler numbers that is

closely related to Bernoulli numbers, Genocchi numbers,

etc. For more information about these polynomials, look

at [1-21] and the references cited therein.

In the present paper, we obtain not only new but also

some interesting identities which are derived from the

fermionic p-adic q-integral over the p-adic numbers field.

The results derived here is written under Sym (5).

2. Symmetric Identities Involving

q-Frobenius-Euler Polynomials

For iw with 1 mod 2iw with 1,2,3,4,5 ,i

by the Eqs. (1.3) and (1.5), we obtain

1 2 3 41 2 3 4 55 4 2 35 4 1 35 4 1 25 3 1 21 2 3 4

1

1 2 3 41 2 3 4 55 4 2 35 4 1 35 4 1 215 3 1 21 2 3 4

0

2lim 1

1

w w w w yw w w w w xw w w w i

tw w w w jw w w w kw w w w hw w w w y q

p

w w w w yw w w w w xw w w w iw w w w jw w w w kNp w w w w hy w w w w y

Ny

e d y

e

t

q

(2.1)

1 151 2 3 4 5

0 0

1 2 3 4 5 1 2 3 4 55 4 2 3 5 4 1 35 4 1 2 5 3 1 2

2lim 1

1

.

Nw pl y w w w w l w y

Nl y

w w w w l w y w w w w w xw w w w i w w w w j tw w w w k w w w w h

qe

Taking

11 1 131 2 4

5 4 2 3 5 4 1 30 0 0 0 5 4 1 2 5 3 1 2

11

2

i j k hww w w

w w w w i w w w w jw w w w k w w w w hi j k h

on the both sides of Eq. (2.1) gives

11 1 131 2 4

5 4 2 3 5 4 1 30 0 0 0 5 4 1 2 5 3 1 2

1 2 3 41 2 3 4 55 4 2 35 4 1 35 4 1 25 3 1 21 2 3 4

1

11

2

i j k hww w w

w w w w i w w w w jw w w w k w w w w hi j k h



pe d y

1 11 1 1 13 51 2 4

0 0 0 0 0 0

1 2 3 4 5 5 4 2 35 4 1 3 5 4 1 2 5 3 1 2

1 2 3 4 5 1 2 3 4 55 4 2 3 5 4 1 35 4 1 2 5 3 1 2

lim 1

Nw ww w w pi j k h y l

Ni j k h l y

w w w w l w y w w w w iw w w w j w w w w k w w w w h

w w w w l w y w w w w w xw w w w i w w w w jw w w w k w w w w h

e

.

t

q

(2.2)

Note that the equation (2.2) is invariant for any

permutation 5.S Hence, we have the following

theorem.

Theorem 1. Let iw with 1 mod 2iw with

1,2,3,4,5 .i Then the following

1 1 1 11 2 3 4

0 0 0 0

5 4 2 3

5 4 1 3

5 4 1 2

5 3 1 2

1 2 3 4 5

1 2 3 4

1 2 3 4 5

5 4 2 3

5 4

11

2

exp([

w w w w

i j k h

i j k h

w w w w i

w w w w j

w w w w k

w w w w h

w w w w l w y

p

w w w w y

w w w w w x

w w w w i

w w w

1 3

5 4 1 2

15 3 1 2 ] )q

w j

w w w w k

w w w w h t d y


holds true for any 5.S

By Eq. (1.2), we easily derive that

1 2 3 4 1 2 3 4 5 5 4 2 3

5 4 1 3 5 4 1 2 5 3 1 2

55

11 2 3 4

5 5 5

2 3 4 1 2 3 4

.

q

q

w w w wq

w w w w y w w w w w x w w w w i

w w w w j w w w w k w w w w h

wy w x i

ww w w w

w w wj k h

w w w

(2.3)

From Eq. (2.1) and (2.3), we obtain

1 2 3 41 2 3 4 55 4 2 35 4 1 35 4 1 25 3 1 21 2 3 4

1

1 2 3 40

1 2 3 4

55

1

5 5

2 3

5

4 1 2 3 4



p

n

pn

w w w w y

n

w w w wq

e d y

w w w w

wy w x i

w

w wj k

w w

wh

w

1 ,!

n

p

td y

n

(2.4)

from which, we have

1 2 3 4

1 2 3 4 5

5 4 2 31 2 3 41

5 4 1 3

5 4 1 2

5 3 1 2

1 2 3 4 1 2 3 4,

5 5 5 5 1 2 3 45

1 2 3 4

2

1

| .

0 .

n

w w w w y

p

q

nw w w wq n q

w w w w

w w w w y

w w w w w x

w w w w id y

w w w w j

w w w w k

w w w w h

w w w w H

w w w ww x i j k h

w w w w

n

(2.5)

Thus, by Theorem 1 and (2.5), we procure the

following theorem.

Theorem 2. For iw with 1 mod 2iw with

1,2,3,4,5 ,i the following

1 2 3 4

1 1 1 11 2 3 4

0 0 0 0

5 4 2 3 5 4 1 3

5 4 1 2 5 3 1 2

1

n

q

w w w w

i j k h

i j k h

w w w w i w w w w j

w w w w k w w w w h

w w w w

1 2 3 4,

5 5 5

51 2 3

5 1 2 3 4

4

|

w w w wn q

w w w w

H

w w ww x i j k

w w w

wh

w

holds true for any 5.S

It is shown by using the definition of [ ]qx that

5 5 5 55

1 2 3 4 1 2 3 4

5

1 2 3 40

4 2 3 4 1 3

4 1 2 3 1 2 5

5 4 2 3 5 4 1 35 4 1 2 5 3 1 2

51 2 3 4

.

n

w w w wq

n mn

q

m q

n m

wq

w w w w i w w w w jm

mw w w w k w w w w h

w w w wq

w w w wy w x i j k h

w w w w

wn

m w w w w

w w w i w w w j

w w w k w w w h

q y w x

(2.6)

Taking 1 2 3 41

w w w w y

pd y on the both sides of

Eq.(2.6) gives

55

1

51 2 3 41

2

5 5

3 4 1 2 3 4

5

1 2 3 40

4 2 3 4 1 3 4 1 2 3 1 25

5 4 2 3 5 4 1 3 5 4 1 2 5 3 1 2

1 2

n

w w w w y

p

w w w wq

n mn

q

m q

n m

wq

m w w w w i w w w w j w w w w k w w w w h

w w

wy w x i

w

wj d y

w

w wk h

w w

wn

m w w w w

w w w i w w w j w w w k w w w h

q

3 45 1

1 2 3 4

5

1 2 3 40

4 2 3 4 1 3 4 1 2 3 1 25

5 4 2 3 5 4 1 3 5 4 1 2 5 3 1 2

1 2 3 451 2 3 4,

2

1

| .

mw w yw w w w

qp

n mn

q

m q

n m

wq

m w w w w i w w w w j w w w w k w w w w h

w w w ww w w w

n q

y w x d y

wn

m w w w w


q

H w x

(2.7)

By the Eq. (2.7), we have

11 1 131 2 4

1 2 3 40 0 0 0

11

2

ww w wn i j k h

qi j k h

w w w w


5 4 2 3 5 4 1 3 5 4 1 2 5 3 1 2

55

1

51 2 3 41

2

5 5

3 4 1 2 3 4

1 2 3 4 50

1 2 3 451 2 3 4,

13 4

0 0

|

1

w w w w i w w w w j w w w w k w w w w h

n

w w w w y

p

w w w wq

nm n m

q qm

w w w ww w w w

n q

w wi j k h

k h

wy w x i

w

wj d y

w

w wk h

w w

nw w w w w

m

H w x

11 11 2

0 0

5 4 2 3 5 4 1 3 5 4 1 2 5 3 1 2

4 2 3 4 1 3 4 1 2 3 1 2

2 4 3 1 3 4 1 2 4 1 2 35

1 2 3 4 50

1 2 3 451 2 3 4,

15,

|

,

w w

i j


m w w w i w w w j w w w k w w w h

n m

wq

nm n m

q qm

w w w ww w w w

n q

wn q

q


nw w w w w

m

H w x

C w

2 3 4, , | ,w w w m(2.8)

where

1 2 3 45,

11 1 131 2 4

0 0 0 0

5 4 2 3 5 4 1 3 5 4 1 2 5 3 1 2

4 2 3 4 1 3 4 1 2 3 1 2

2 4 3 1 3 4 1 2 4 1 2 35

, , , |

1

.

wn q

ww w wi j k h

i j k h


m w w w i w w w j w w w k w w w h

n m

wq

C w w w w m

q


(2.9)

Consequently, by (2.9), we get the following theorem.

Theorem 3. Let iw with 1 mod 2iw with

1,2,3,4,5 .i Then the following expression

51 2 3 40

1 2

3 451 2 3 4

,

1 2 3 45,

|

, , , |

n m n m

qqm

w w

w w

w w w wn q

wn q

nw w w w w

m

H w x

C w w w w m

holds true for some 5.S

3. Conclusion

We have derived some new interesting identities of

q-Frobenius-Euler polynomials. We also showed that

these symmetric identities are written by symmetric group

of degree five.

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