Entire Functions and Their Derivatives Share Two Finite Sets

• Journal title : Kyungpook mathematical journal
• Volume 49, Issue 3,  2009, pp.473-481
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2009.49.3.473
Title & Authors
Entire Functions and Their Derivatives Share Two Finite Sets
Meng, Chao; Hu, Pei-Chu;

Abstract
In this paper, we study the uniqueness of entire functions and prove the following theorem. Let n($\small{{\geq}}$ 5), k be positive integers, and let $\small{S_1}$ = {z : $\small{z^n}$ = 1}, $\small{S_2}$ = {$\small{a_1}$, $\small{a_2}$, $\small{{\cdots}}$, $\small{a_m}$}, where $\small{a_1}$, $\small{a_2}$, $\small{{\cdots}}$, $\small{a_m}$ are distinct nonzero constants. If two non-constant entire functions f and g satisfy $\small{E_f(S_1,2)}$ = $\small{E_g(S_1,2)}$ and $\small{E_{f^{(k)}}(S_2,{\infty})}$ = $\small{E_{g^{(k)}}(S_2,{\infty})}$, then one of the following cases must occur: (1) f = tg, {$\small{a_1}$, $\small{a_2}$, $\small{{\cdots}}$, $\small{a_m}$} = t{$\small{a_1}$, $\small{a_2}$, $\small{{\cdots}}$, $\small{a_m}$}, where t is a constant satisfying $\small{t^n}$ = 1; (2) f(z) = $\small{de^{cz}}$, g(z) = $\small{\frac{t}{d}e^{-cz}}$, {$\small{a_1}$, $\small{a_2}$, $\small{{\cdots}}$, $\small{a_m}$} = $\small{(-1)^kc^{2k}t\{\frac{1}{a_1},{\cdots},\frac{1}{a_m}\}}$, where t, c, d are nonzero constants and $\small{t^n}$ = 1. The results in this paper improve the result given by Fang (M.L. Fang, Entire functions and their derivatives share two finite sets, Bull. Malaysian Math. Sc. Soc. 24(2001), 7-16).
Keywords
entire function;share set;uniqueness;
Language
English
Cited by
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