The Periodicity of Entire Functions with Finite Order

Kit C Chan,Guoqiang Dang,Zhiguo Ren

doi:10.1155/2021/9995695

Kit C Chan, Guoqiang Dang + Show 1 more

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https://doi.org/10.1155/2021/9995695

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Abstract

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

Highlights

Theorem 1 shows that Yang’s conjecture is true when k 1. e transcendental entire function f2 cannot replaced by f in eorem 1 [2], due to a counterexample which had been presented by Liu and Yu which shows that f(z) ez + z is not a periodic function, but f(k)(z) is a periodic function [2]. ey depicted that the function f2 can be replaced by fn in eorem 1 provided that n ≥ 3 [2]
In eorem 1, if we restrict f 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 f has an explicit expression
F(z) is a transcendental entire function with finite order and d is its Picard exceptional value; f(z) d + eP(z), where P(z) is a polynomial of degP(z) ≥ 1, and T(r, P) S(r, f)

Summary

Introduction

If f is a transcendental entire function with a nonzero Picard exceptional value, Liu and Yu [2] proved the following theorem in 2019. In eorem 1, if we restrict f 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 f has an explicit expression.

Results

Conclusion