Uniqueness of first derivatives and differences in meromorphic functions and the characterization of entire function periodicity

TL;DR

This paper investigates two intertwined problems in Nevanlinna theory for difference operators: a uniqueness problem for $f^{(1)}$ and $\Delta_c f$ under shared values, and periodicity criteria for transcendental entire functions. For meromorphic $f$ with finite hyper-order $\rho_1(f)$, it proves that if $f^{(1)}$ and $\Delta_c f$ share two finite values $a_1,a_2$ and $\infty$ CM, then $f^{(1)}\equiv \Delta_c f$; an analogous result holds for entire $f$ sharing two finite values CM, thereby resolving Qi et al.'s open problem in the finite hyper-order setting. The paper also establishes sufficient conditions for the periodicity of transcendental entire functions: if $\frac{f^{(k)}}{f^n(f^m-1)}$ is periodic with period $c$, and either $f(z)=e^{P(z)}$ with non-constant $P$, or $f$ has a non-zero Picard exceptional value with $\rho_1(f)<\infty$, or $\frac{f^{(k+1)}}{f^n(f^m-1)}$ is periodic, then $f$ must be periodic. Collectively, these results advance the understanding of how difference operators constrain Meromorphic functions and contribute to the Yang conjecture lineage and Wei et al.'s questions on periodicity.

Abstract

The objective of the paper is twofold. The first objective is to study the uniqueness problem of meromorphic function $f(z)$ when $f^{(1)}(z)$ shares two distinct finite values $a_1$, $a_2$ and $\infty$ CM with $Δ_cf(z)$. In this context, we provide a result that resolves the open problem posed by Qi et al. [Comput. Methods Funct. Theory, 18 (2018), 567-582] for the case when hyper order of the function is less than $\infty$. The second objective is to establish sufficient conditions for the periodicity of transcendental entire functions. In this direction, we obtain a result that affirms the question raised by Wei et al. [Anal. Math., 47 (2021), 695-708.]

Theorems & Definitions (13)

• Theorem 2.1
• Corollary 2.1
• Remark 2.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• proof : Proof of Theorem \ref{['t1']}