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 [7–9]. 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 3–6, 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.