Bulletin of the
Korean Mathematical Society
BKMS

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

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Bull. Korean Math. Soc. 2020; 57(6): 1541-1565

Online first article November 2, 2020      Printed November 30, 2020

https://doi.org/10.4134/BKMS.b200051

Copyright © The Korean Mathematical Society.

Zero distribution of some delay-differential polynomials

Ilpo Laine, Zinelaabidine Latreuch

University of Eastern Finland; University of Mostaganem

Abstract

Let $f$ be a meromorphic function of finite order $\rho$ with few poles in the sense $S_{\lambda}(r,f):=O(r^{\lambda +\varepsilon})+S(r,f)$, where $\lambda <\rho$ and $\varepsilon \in (0,\rho-\lambda)$, and let $g(f):=\sum_{j=1}^{k}b_{j}(z)f^{(k_j)}(z+c_{j})$ be a linear delay-differential polynomial of $f$ with small meromorphic coefficients $b_{j}$ in the sense $S_{\lambda}(r,f)$. The zero distribution of $f^{n}(g(f))^{s}-b_{0}$ is considered in this paper, where $b_0$ is a small function in the sense $S_{\lambda}(r,f)$.

Keywords: Meromorphic functions, delay-differential polynomial, shifts, zero distribution

MSC numbers: Primary 30D35

Supported by: The first author has been partially supported by The Academy of Finland project no. 286877. The second author has been supported by the Directorate General for Scientific Research and Technological Development (DGRSDT), Algeria