QUASIPERIODICITY OF TRANSCENDENTAL MEROMORPHIC FUNCTIONS*

(刘新玲 )

Department of Mathematics, Nanchang University, Nanchang 330031, China;

Department of Physics and Mathematics, University of Eastern Finland,P. O. Box 111, 80101, Joensuu, Finland E-mail: liuxinling@ncu.edu.cn

Kai LIU (刘凯)

Department of Mathematics, Nanchang University, Nanchang 330031, China E-mail: liukai@ncu.edu.cn

Risto KORHONEN†

Department of Physics and Mathematics, University of Eastern Finland,P. O. Box 111, 80101, Joensuu, Finland E-mail: risto.korhonen@uef.fi

Galina FILIPUK

Institute of Mathematics, Faculty of Mathematics, Informatics and Mechanics,University of Warsaw, Banacha 2, 02-097, Warsaw, Poland E-mail: G.filipuk@uw.edu.pl

Abstract This paper is devoted to considering the quasiperiodicity of complex differential polynomials, complex difference polynomials and complex delay-differential polynomials of certain types, and to studying the similarities and differences of quasiperiodicity compared to the corresponding properties of periodicity.

Key words quasiperiodicity; meromorphic functions; complex delay-differential polynomials; Nevanlinna theory

1 Introduction

Letfbe a transcendental meromorphic function in the complex plane.The functionf(z)is a periodic meromorphic function with periodc, iff(z+c) =f(z) for allz ∈C andcis a non-zero constant.Ozawa [15] proved the existence of periodic entire functions of any positive order and of hyper-order at least one.

Letφ(z) andψ(z) be two polynomials such thatφ(z)/≡0,1 andψ(z)/≡0.We present two definitions of quasiperiodic functions below which describe theφ-time quasiperiodicity andψ-plus quasiperiodicity for meromorphic functions.

Definition 1.1Iff(z) satisfiesf(z+c)=φ(z)f(z), thenf(z) is aφ-time quasiperiodic function.

Ifφ(z) reduces to a constantqand (q/= 0,1), thenf(z) is sometimes called geometric quasiperiodic function.Actually,ifc=1 and the polynomialφ(z)has the following presentation

whereλ/=0,αk(k=1,2,···,n)are complex numbers,then theφ-time quasiperiodic functionsf(z) can be written as

whereg(z) is an arbitrary periodic function with period 1 and Γ(z) is the gamma function,see [5, pp115-116].We also know that any non-constant rational function cannot be aφ-time quasiperiodic function.The transcendentalφ-time quasiperiodic functionsf(z) are of orderρ(f)≥1.

Definition 1.2Iff(z)satisfiesf(z+c)=f(z)+ψ(z),thenf(z)is aψ-plus quasiperiodic function or quasiperiodic function modψ(z).

Ifψ(z) is a non-zero constant, thenf(z) is also sometimes called arithmetic quasiperiodic function.Theψ-plus quasiperiodic functions can be written asf(z)=π(z)+ς(z),whereπ(z)is a periodic function with periodcandς(z) is a polynomial that satisfiesς(z+c)-ς(z)=ψ(z).Thus, any non-constant polynomials areψ-plus quasiperiodic functions.For transcendentalψ-plus quasiperiodic functionsf(z), we also see thatρ(f)≥1 for the reason thatρ(π)≥1 wheneverπ(z) is a non-constant periodic function with periodc.

The periodicity of transcendental meromorphic functions has been considered from different aspects, such as the periodicity with uniqueness theory related to value sharing, composite functions, complex differential equations, or complex difference equations.Some results and references on these topics have been collected recently in [14], where the latest considerations of the periodicity, related to Yang’s Conjecture and its variations, are also included, see also[8, 10-13, 18].Recall Yang’s Conjecture as follows, see [7, 17].

Yang’s ConjectureLetfbe a transcendental entire function andkbe a positive integer.Ifff(k)is a periodic function, thenfis also a periodic function.

The motivation of the paper is to address the question posed by G.Filipuk during the first author’s PhD dissertation defence at the University of Eastern Finland, which is how to consider Yang’s Conjecture and its variations in the context of quasiperiodicity?

The main results of the paper involve considering the above two notions of quasiperiodicity for composite functions, complex differential polynomials, complex difference polynomials and complex delay-differential polynomials of certain types.The paper is organized as follows.In Section 2, we discuss the quasiperiodicity of composite functions.In Section 3, we consider the corresponding versions for quasiperiodicity based on the generalized Yang’s Conjecture and its variations.Section 4 examines the quasiperiodicity of delay versions of Yang’s Conjecture,while Section 5 explores the quasiperiodicity of delay-differential versions of Yang’s Conjecture.

2 Quasiperiodicity of Composite Functions

We assume thatP(z) is a non-constant polynomial andf(z) is an entire function.Concerning the periodicity ofP(f(z)), R´enyi and R´enyi [16, Theorem 2] obtained that iff(z) is not periodic, thenP(f(z)) cannot be a periodic function.On the quasiperiodicity ofP(f(z)),we obtain the following result.

Theorem 2.1Letf(z) be a transcendental entire function andP(z) be a non-constant polynomial.If deg(P(z))≥2, thenP(f(z)) is not aψ-plus quasiperiodic function.

ProofIf deg(P(z))=2, we assume thatP(z)=a2z2+a1z+a0, wherea2/=0.We will prove that there are no transcendental entire solutions, except ifψ(z)≡0, for the functional equation

Sinceψ(z) is a non-zero polynomial, thenψ(z+(m-1)c)+···+ψ(z+c)+ψ(z)/≡0.Then we have eitherF(z)2+ψ(z+(m-1)c)+···+ψ(z+c)+ψ(z) have no zeros or the zeros ofF(z)2+ψ(z+(m-1)c)+···+ψ(z+c)+ψ(z) are multiple for allm ≥1, and we obtain a contradiction with the second main theorem of Nevanlinna theory for small functions, see [19].

If deg(P(z)) = 3, we assume thatP(z) =a3z3+a2z2+a1z+a0.In the proof of [18,Theorem 1.1], Wei, Liu and Liu obtained

has no transcendental entire solutions except whenψ(z)≡0.If deg(P(z))≥4, we assume thatP(f(z+c)) =P(f(z))+ψ(z).By [2, Theorem 3.4], we seeψ(z) must be a constant,and then the proof of [18, Theorem 1.1] shows thatP(f(z+c)) =P(f(z))+dhas also no transcendental entire solutions except whend ≡0.Summarizing the above results, we have thatP(f(z)) cannot be aψ-plus quasiperiodic function.□

Remark 2.2The above method for deg(P(z))=2 is also valid for transcendental meromorphic functions, however, the case deg(P(z))≥3 remains open for transcendental meromorphic functions.

Recall the basic notation and fundamental results of Nevanlinna theory(see,e.g.,[1,4,20]),such as the proximity functionm(r,f),the counting functionN(r,f),the characteristic functionT(r,f), the orderρ(f), and the hyper-orderρ2(f).

Theorem 2.3LetP(z)be a non-constant polynomial.If deg(P(z))≥2,then theφ-time quasiperiodicity ofP(f(z)) can be stated as follows:

(i) If deg(P(z))=2,P(z) has two distinct zeros andf(z) is a meromorphic function, thenP(f(z)) cannot be aφ-time quasiperiodic function except whenφn=1 forn=2,3.

Remark 2.4(1)IfP(z)=zandf(z)=ez+z,then ez+2πi+z+2πi=ez+z+2πi.Hence,ez+zis 2πi-plus quasiperiodic function, wherec=2πi.This example shows that Theorem 2.1 is not true for deg(P(z))=1.

(2) IfP(z) has only one zero, thenP(f(z)) can be az2-time quasiperiodic function.For instance, (Γ(z+1))2=z2(Γ(z))2.This means the condition thatP(z) has two distinct zeros cannot be removed in (i) of Theorem 2.3.

(3) Ifφ ≡-1 in (2.4), then we consider the meromorphic solutions for Fermat difference equation

Conjecture 2.5If deg(P(z))≥2, thenf(P(z)) is notφ-time quasiperiodic, wheref(z)is a transcendental meromorphic function andP(z) is a polynomial.

3 Generalized Yang’s Conjecture for Quasiperiodicity

Let us recall the generalized Yang’s Conjecture in[10],where the casen=1 is called Yang’s Conjecture in [17].

Generalized Yang’s ConjectureLetf(z) be a transcendental entire function andn,kbe positive integers.Iff(z)nf(k)(z)is a periodic function, thenf(z)is also a periodic function.

Liu, Wei and Yu [10] obtained partial answers to the generalized Yang’s Conjecture as follows by giving additional conditions.

Theorem ALetf(z) be a transcendental entire function andn,kbe positive integers.Assume thatf(z)nf(k)(z)is a periodic function with periodc.If one of the following conditions is satisfied

(i)k=1;

(ii)f(z)=eh(z), whereh(z) is a non-constant polynomial;

(iii)f(z) has a non-zero Picard exceptional value andf(z) is of finite order;

(iv)f(z)nf(k+1)(z) is a periodic function with periodc;thenf(z) is a periodic function.

φ-time Quasiperiodic Version of Generalized Yang’s ConjectureLetfbe a transcendental entire function andn,kbe positive integers.Iff(z)nf(k)(z)is aφ-time quasiperiodic function, thenfis also aφ-time quasiperiodic function.

The aboveφ-time quasiperiodic version of Yang’s conjecture can be considered by similar methods which have been used in considering Yang’s conjecture.Most results on the periodicity in Yang’s conjecture can be improved directly, and we will not give these considerations.We mainly consider theψ-plus quasiperiodicity and obtain

Theorem 3.1Letfbe a transcendental entire function withρ2(f)＜1 andn,kbe positive integers.

(i) Ifn ≥3, thenf(z)nf(k)(z) cannot be aψ-plus quasiperiodic function with periodc.

(ii) Ifn ≤2, thenf(z)nf′(z) cannot be aψ-plus quasiperiodic function with periodc.

Proof(i) Iffnf(k)is aψ-plus quasiperiodic function with periodc, we can assume that

We will affirm thatψs(z)≡0 (s=0,1,2,···,q-1), whereψ0(z)=ψ(z).Otherwise, ifψs(z)are mutually distinct,using the second main theorem of Nevanlinna theory[19],[3,Lemma 8.3]and a basic inequality

Hence,we have(q-1)n ≤2q,however ifn ≥3,by taking a large enoughqwe get a contradiction.Hence,at least two of the functionsψs(z)are identically the same,thus,ψ(z)≡0 for the reason thatψ(z) is a polynomial.

(ii) Assume thatfnf′is aψ-plus quasiperiodic function with periodc, then

which is impossible forn=1 orn=2.□

Remark 3.2(1)The question of whetherfnf(k)(n ≤2,k/=1)can be aψ-plus quasiperiodic function with periodcor not remains open.

(2) We need the conditionρ2(f)＜1 to get the relationshipT(r,f(z+sc))=T(r,f(z))+S(r,f).Here, we conjecture

from the equation (3.1).

The following theorem can be seen as another version of Theorem 3.1 where we allownto have negative integer values.

Theorem 3.3Letfbe a transcendental entire function withρ2(f)＜1 andn,kbe positive integers.

Here, by a difference analogue of the logarithmic derivative lemma for transcendental entire functionfwithρ2(f)＜1 (see [3, Theorem 5.1]) and Lemma 3.4, we get

4 Delay Yang’s Conjecture for Quasiperiodicity

Liu and Korhonen[12,Theorem 1.3]considered the delay Yang’s Conjecture for periodicity as follows.

Theorem BLetf(z) be a transcendental entire function withρ2(f)＜1 andn ≥2 be a positive integer.Iff(z)nf(z+η) is a periodic function with periodc, thenf(z) is a periodic function with period (n+1)c.

We now state the version below on delay Yang’s Conjecture for quasiperiodicity.

Theorem 4.1Letf(z) be a transcendental entire function withρ2(f)＜1.

(1) Ifn ≥2 andf(z)nf(z+η) is aφ-time quasiperiodic function with periodc, thenf(z+c)=G(z)f(z), whereG(z) is a rational function satisfyingG(z)nG(z+η)=φ(z).

(2) Ifn ≥4, thenf(z)nf(z+η) is not aψ-plus quasiperiodic function with periodc.

Proof(1) Assume thatf(z)nf(z+η) is aφ-time quasiperiodic function with periodc.Then

which contradicts withn ≥2.Thus,G(z)should be a rational function and satisfyG(z)nG(z+η)=φ(z).

(2) Assume thatf(z)nf(z+η) is aψ-plus quasiperiodic function with periodη.Then

thusf(z) is not aφ-time quasiperiodic function with period 1, howeverf(z+1)nf(z+2) is aφ-time quasiperiodic function with period 1 by

(ii) Using [9, Lemma 2.5] and similar proofs as above, Theorem 4.1 (2) is true forn ≥8 whenfis a transcendental meromorphic function withρ2(f)＜1.

5 Delay-Differential Yang’s Conjecture for Quasiperiodicity

Liu and Korhonen[12,Theorem 1.5]also presented a version on the delay-differential Yang’s Conjecture for periodicity, which can be stated as follows.

Theorem CLetf(z) be a transcendental entire function withρ2(f)＜1 andn ≥4 be a positive integer.If[f(z)nf(z+η)](k)is a periodic function with periodc,thenf(z)is a periodic function with period (n+1)c, wherekis a positive integer.

Finally, we also provide a result on the delay-differential Yang’s Conjecture for quasiperiodicity.

Theorem 5.1Letf(z) be a transcendental entire function withρ2(f)＜1.Ifn ≥2,then [f(z)nf(z+c)](k)cannot be aψ-plus quasiperiodic function with periodη.

ProofIf [f(z)nf(z+c)](k)is aψ-plus quasiperiodic function with periodη, we can assume that

Hence, we have (q- 1)(n+ 1)≤2q, however ifn ≥2, there exists a suitableqto get a contradiction.□

Remark 5.2Iff(z)is a transcendental meromorphic function withρ2(f)＜1 in Theorem 5.1, by using[9, Lemma 2.5]and the similar proofs as above, we can get(q-1)(n-1)T(r,f)≤(2q+2)T(r,f)+S(r,f).Hence, Theorem 5.1 is true whenn ≥4 andf(z) is a transcendental meromorphic function withρ2(f)＜1.

Conflict of InterestThe authors declare no conflict of interest.