Journal of Mathematics

Journal of Mathematics / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 9995695 | https://doi.org/10.1155/2021/9995695

Zhiguo Ren, Guoqiang Dang, "The Periodicity of Entire Functions with Finite Order", Journal of Mathematics, vol. 2021, Article ID 9995695, 8 pages, 2021. https://doi.org/10.1155/2021/9995695

The Periodicity of Entire Functions with Finite Order

Academic Editor: kit C. Chan
Received04 Mar 2021
Revised06 Apr 2021
Accepted09 Apr 2021
Published06 May 2021

Abstract

This paper is concerned with the periodicity of entire functions with finite growth order, and some sufficient conditions are given. Let is a transcendental entire function with finite growth order, zero is a Picard exceptional value of , and a given differential monomial of is periodic, then is also periodic. We are also interested in finding the following: let is a transcendental entire function with finite growth order, is a Picard exceptional value of and is a periodic function, then is also a periodic function. These results extend Yang’s conjecture.

1. Introduction

The aim of this paper is to bring together and generalize recent research work by Wang and Hu [1], Liu and Yu [2], Deng and Yang [3], Liu et al. [4], Lü and Zhang [5], and Liu et al. [6] which is related to Yang’s conjecture [1, 2].

Conjecture 1. (Yang’s conjecture [1, 2]). Let be a transcendental entire function and be a positive integer. If is a periodic function, then is also a periodic function.
We assume that the readers are familiar with Nevanlinna’s theory [79]. Denote by any quantify satisfying , , outside of a possible exceptional set of finite logarithmic measure. For a meromorphic function , define its shift by and its difference operators byFor a nonconstant meromorphic function , the order of is defined byWang and Hu [1] proved the following theorem.

Theorem 1. (see [1, Theorem 1]). Let be a transcendental entire function and let be a positive integer. If is a periodic function, then is also a periodic function.
If is a transcendental entire function with a nonzero Picard exceptional value, Liu and Yu [2] proved the following theorem in 2019.

Theorem 2. (see [2, Theorem 1.1]). Let be a transcendental entire function with a nonzero Picard exceptional value and let be a positive integer. If is a periodic function, then is also a periodic function.
In fact, Theorem 1 shows that Yang’s conjecture is true when . The transcendental entire function cannot replaced by in Theorem 1 [2], due to a counterexample which had been presented by Liu and Yu which shows that is not a periodic function, but is a periodic function [2]. They also depicted that the function can be replaced by in Theorem 1 provided that [2]. In Theorem 2, Liu and Yu [2, Theorem 1.1] investigated the periodicity of a transcendental entire function with a nonzero Picard exceptional. Furthermore, in Theorem 1, if we restrict be a transcendental entire function with finite order and zero is a Picard exceptional value, we can obtain the following more refined theorem which shows that has an explicit expression.

Theorem 3. Let be a transcendental entire function with finite order, zero be a Picard exceptional value of , and be a positive integer. If is a periodic function with a period of , then is also a periodic function with a period of ; furthermore, , where are constants and .
If we replace by in Theorem 3, we can obtain the following theorem.

Theorem 4. Let be a transcendental entire function with finite order and zero be a Picard exceptional value of , and let be a positive integer. If is a periodic function with a period of , then is also a periodic function with a period of ; furthermore, , where are constants and .
In 2020, under the conditions of Theorem 2, if the condition “ with a nonzero Picard exceptional value” replaced by “zero be a Picard exceptional value of ,” Deng and Yang [3] proved the following theorem.

Theorem 5. (see [3, Theorem 3]). Let be a transcendental entire function with finite order and zero be a Picard exceptional value of and let be a positive integer. If is a periodic function with a periodic of , then is also a periodic function; furthermore, , where are constants, , and is with a period of .
We generalize Theorem 5 and obtain the following result.

Theorem 6. Let be a transcendental entire function with finite order and zero be a Picard exceptional value of and let be a positive integer. If is a periodic function with a period of , then is also a periodic function with a period of ; furthermore, , where are constants and .
The following generalized Yang’s conjecture has been considered by Liu et al. in 2020 [4].

Conjecture 2. (generalized Yang’s conjecture [4]). Let be a transcendental entire function and be positive integers. If is a periodic function, then is also a periodic function.
If is a transcendental entire function, Liu et al. [4] proved that the aforementioned generalized Yang’s conjecture is true under the conditions that if or if , where is a nonconstant polynomial, or if has a nonzero Picard exceptional value and is of finite order. In 2021, Liu et al. [6] proved that the generalized Yang’s conjecture is true for meromorphic functions in the case of , while for , the conjecture is true under some certain conditions even if is a negative integer value. Baker [10] proved that if is a nonconstant entire function and is a polynomial with , then is not a periodic function. Contrarily, when is periodic, where is a nonconstant polynomial, it is interesting to discuss the periodicity of . Let be a transcendental entire function, be a positive integer, be constants, and ; Liu and Yu [2, Theorem 1.5] proved that if is a periodic function, then is also a periodic function. Staring form Theorems 36, we have a question that whether is also periodic if we change the conditions , , and into a differential polynomial of . Furthermore, let is a transcendental entire function with and are two positive integers, and , Liu and Yu ([2], Theorem 1.7) also proved that if is a periodic function, , then is also a periodic function, but the exact formula of is still unobtained [2]. In order to construct the explicit expression of , it is necessary to put some restrictions of . In fact, we obtain the following theorem.

Theorem 7. Let be a transcendental entire function with finite order and zero be a Picard exceptional value of , and let be a positive integer. If is a periodic function with a period of , then is also periodic with a period of ; furthermore, , where are constants, , and .
Noting that if we set and , Theorem 7 will reduce to Theorem 6.
In 2020, Deng and Yang [3] studied the difference analogue of Yang’s conjecture and obtained the following theorem.

Theorem 8. (see [3, Theorem 4]). Let be a transcendental entire function with finite order and be a Picard exceptional value of , and be a periodic function; then, is also a periodic function.
In Theorem 3, if we replace into , we can obtain the following theorem.

Theorem 9. Let be a transcendental entire function with finite order and be a Picard exceptional value of , and be a periodic function; then, is also a periodic function.
Theorems 8 and 9 partly show the internal relation of periodicity between a transcendental entire function and differences; in a more general setting, we can put the following theorem.

Theorem 10. Let be a transcendental entire function with finite order and be a Picard exceptional value of , and be a periodic function; then is also a periodic function.

2. Lemma

Lemma 1. (see [11]). Let be a transcendental entire function with finite order and be a nonzero constant; then, .

Lemma 2. (see [7]). Let be meromorphic functions and be nonconstant; , and for any ,where is a set of finite measure; then, .

3. Proofs of Theorems

3.1. Proof of Theorem 3

Proof. By Theorem 1, it follows easily that is periodic. We only need to show that , where and .
Suppose that , where is a polynomial, , . Differentiate times, , where is a differential polynomial of , . Otherwise, if , then ; then, is a polynomial, but it is a contradiction with which is a transcendental entire function.
Since is a periodic function with a period of , ; consequently,and in this way,In the event of , then . By Lemma 1, we have , . Butwhich is impossible. Therefore, .
It follows that ; then, , , ; consequently, . Hence, .
This finishes the proof.

3.2. Proof of Theorem 4

Proof. According to the proof of Theorem 3, we have , where is a differential polynomial of , , and .
Because is a periodic function with a period of , thenIt follows thatand accordingly,In the event of , . By Lemma 1, we have , . Therefore,which is impossible. Hence, .
Assuming that . Then, , , and ; consequently, . Hence, ; then, is a periodic function with a period of .
This finishes the proof.

3.3. Proof of Theorem 6

Proof. Suppose that , where is a polynomial, and and . Then, , where is a differential polynomial of and . Otherwise, if , then ; therefore, is a polynomial, which is contradicting with which is a transcendental entire function with finite order.
Since is a periodic function with a period of , thenand in this way,Taking the virtue of Lemma 1, we have , . Therefore,In the event of , . Butwhich is impossible. Hence, .
Assuming that . So, , , and by (11), we have ; therefore, . Hence, ; consequently, is a periodic function with a period of .
This finishes the proof.

3.4. Proof of Theorem 7

Proof. Suppose that , where is a polynomial, and and . Differentiate times; then , where is a differential polynomial of and . Otherwise, if , then ; then, is a polynomial, but it is contradicting with which is a transcendental entire function with finite order. Therefore,where is a differential polynomial of . Since is a periodic function with a period of , then , andTherefore,In the event of , . By Lemma 1, we have , . Butwhich is impossible. Hence, .
Assuming that . So, , , and by (16), we have , where , so . Therefore, , and hence is a periodic function with a period of .
This finishes the proof.

3.5. Proof of Theorem 9

Proof. Let be a transcendental entire function with finite order and be a Picard exceptional value of ; then, , where is a polynomial of , and . By simple computation, we have(i)Case 1. If .Case 1.1. If , then ; therefore, is a periodic function with a period of .Case 1.2. If . Assuming that is a period of , then  Case 1.2.1. .  By and (20), we have  Consequently,  Furthermore,  Therefore, we must have  It follows that  If not, by Lemmas 1 and 2, combining (25) and (26), we have  which is impossible.   Case 1.2.1.1. .   Assuming that . By computation, we have . Hence, , , are polynomials with degree no more than ; by (25) and Lemma 2, we have , or , or , but it is impossible.   Case 1.2.1.2. .   By and (25), we have . By Lemma 2, we have , or , or ; therefore, is a function of a period of , or , or . Since , by (26), ; furthermore , or , or , and hence is a periodic function with a period of or or .  Case 1.2.2. .  By and (20), we have  Consequently,  Therefore, we must have  It follows that  If not, by Lemmas 1 and 2, combining (31) and (32), we have  which is impossible.   Case 1.2.2.1. .   Assuming that . By computation, we have . Hence, , are polynomials with degree no more than ; by (31) and Lemma 2, we have , or , but it is impossible.   Case 1.2.2.2. .   By and (31), we have . By Lemma 2, we have , or ; therefore, is a function of a period of . If , by (32), we have , and it follows that , or , and hence is a periodic function with a period of .(ii)Case 2. If .Case 2.1. If , then ; therefore, is a periodic function with a period of .Case 2.2. If . Assuming that is a period of , then  Case 2.2.1. .  By and (34), we have  Consequently,  Furthermore,  Therefore, we must have   Case 2.2.1.1. .   Assuming that . By computation, we have . Hence, , , are polynomials with degree no more than ; by (38) and Lemma 2, we have , or , or , but it is impossible.   Case 2.2.1.2. .   By and (38), we have . By Lemma 2, we have , or , or ; therefore, is a function of a period of , or , or .  Case 2.2.2. .  By and (34), we have  Consequently,  It follows that   Case 2.2.2.1. .   Assuming that . By computation, we have . Hence, , are polynomials with degree no more than ; by (41) and Lemma 2, we have , or , but it is impossible.   Case 2.2.2.2. .   By and (41), we have . By Lemma 2, we have , or ; therefore, is a function of a period of .   This finishes the proof.

3.6. Proof of Theorem 10

Proof. is a transcendental entire function with finite order and is its Picard exceptional value; then, , where is a polynomial of , and . Then,(i)Case 1. If .We assert that is a periodic function with a period of . If is odd, by (15), we haveBy Lemma 2, for each , we have . Therefore, ; then, , and hence is a periodic function with a period of . If is even, it is easy to prove that is also periodic.(ii)Case 2. If . Assuming that is a periodic of , thenCase 2.1. . By and (15) and (16), Furthermore, It follows that