Kai Liu

Bio: Kai Liu is an academic researcher from Nanchang University. The author has contributed to research in topics: Value (mathematics) & Conjecture. The author has an hindex of 1, co-authored 1 publications receiving 7 citations.

A note on the periodicity of entire functions

Abstract: We give some sufficient conditions for the periodicity of entire functions based on a conjecture of C. C. Yang, using the concepts of value sharing, unique polynomial of entire functions and Picard exceptional value.

On the periodicity of transcendental entire functions

Abstract: According to a conjecture by Yang, if $f(z)f^{(k)}(z)$ is a periodic function, where $f(z)$ is a transcendental entire function and $k$ is a positive integer, then $f(z)$ is also a periodic function. We propose related questions, which can be viewed as difference or differential-difference versions of Yang’s conjecture. We consider the periodicity of a transcendental entire function $f(z)$ when differential, difference or differential-difference polynomials in $f(z)$ are periodic. For instance, we show that if $f(z)^{n}f(z+\\unicode[STIX]{x1D702})$ is a periodic function with period $c$, then $f(z)$ is also a periodic function with period $(n+1)c$, where $f(z)$ is a transcendental entire function of hyper-order $\\unicode[STIX]{x1D70C}_{2}(f)<1$ and $n\\geq 2$ is an integer.

Generalized yang's conjecture on the periodicity of entire functions

Abstract: On the periodicity of transcendental entire functions, Yang’s Conjecture is proposed in [6, 13]. In the paper, we mainly consider and obtain partial results on a general version of Yang’s Conjecture, namely, if f(z)nf (k)(z) is a periodic function, then f(z) is also a periodic function. We also prove that if f(z)n+f (k)(z) is a periodic function with additional assumptions, then f(z) is also a periodic function, where n, k are positive integers.

On the Periodicity of Entire Functions

Abstract: The purpose of this paper is mainly to prove that if f is a transcendental entire function of hyper-order strictly less than 1 and $$f(z)^{n}+a_{1}f'(z)+\cdots +a_{k}f^{(k)}(z)$$ is a periodic function, then f(z) is also a periodic function, where n, k are positive integers, and $$a_{1},\cdots ,a_{k}$$ are constants. Meanwhile, we offer a partial answer to Yang’s Conjecture, theses results extend some previous related theorems.

Variations on a Conjecture of C. C. Yang Concerning Periodicity

Abstract: The generalized Yang’s Conjecture states that if, given an entire function f(z) and positive integers n and k, $$f(z)^nf^{(k)}(z)$$ is a periodic function, then f(z) is also a periodic function. In this paper, it is shown that the generalized Yang’s conjecture is true for meromorphic functions in the case $$k=1$$ . When $$k\ge 2$$ the conjecture is shown to be true under certain conditions even if n is allowed to have negative integer values.

The Periodicity on a Transcendental Entire Function with its Differential Polynomials

Abstract: We discuss the relationship on the periodicity of a transcendental entire function with its differential polynomials. For example, we obtain that if f is a transcendental entire function, k is a non-negative integer and if (anfn + ⋯ + a1f)(k) is a periodic function, then f is also a periodic function, where a1, … an (≠ 0) are constants. Our results are related to Yang’s Conjecture on the periodicity of transcendental entire functions.