Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2021; 58(4): 915-937

Online first article June 7, 2021      Printed July 31, 2021

https://doi.org/10.4134/BKMS.b200670

Copyright © The Korean Mathematical Society.

A result on an open problem of L\"{u}, Li and Yang

Sujoy Majumder, Somnath Saha

Raiganj University; P.O.- Daulatpur, Dist.- Dakshin Dinajpur

Abstract

In this paper we deal with the open problem posed by L\"{u}, Li and Yang \cite{11a}. In fact, we prove the following result: Let $f(z)$ be a transcendental meromorphic function of finite order having finitely many poles, $c_{1},c_{2},\ldots, c_{n}\in\mathbb{C}\setminus\{0\}$ and $k, n\in\mathbb{N}$. Suppose $f^{n}(z)$, $f(z+c_{1})f(z+c_{2})\cdots f(z+c_{n})$ share $0$ CM and $f^{n}(z)-Q_{1}(z)$, $(f(z+c_{1})f(z+c_{2})\cdots f(z+c_{n}))^{(k)}-Q_{2}(z)$ share $(0,1)$, where $Q_{1}(z)$ and $Q_{2}(z)$ are non-zero polynomials. If $n\geq k+1$, then $(f(z+c_{1})f(z+c_{2})\cdots f(z+c_{n}))^{(k)}\equiv \frac{Q_{2}(z)}{Q_{1}(z)}f^{n}(z)$. Furthermore, if $Q_{1}(z)\equiv Q_{2}(z)$, then $f(z)=c\;e^{\frac{\lambda}{n}z}$, where $c, \lambda\in\mathbb{C}\setminus\{0\}$ such that $e^{\lambda (c_{1}+c_{2}+\cdots+c_{n})}=1$ and $\lambda^{k}=1$. Also we exhibit some examples to show that the conditions of our result are the best possible.

Keywords: Meromorphic function, derivative, small function

MSC numbers: 30D35