Growth of entire functions with some univalent Gelfond-Leontev derivatives
type: Item
instance of: scholarly article
title: Growth of entire functions with some univalent Gelfond-Leontev derivatives
zbMATH Open document ID: 0664.30013
DOI: 10.1155/S0161171289000037
author: S. H. Smith
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/46348
review text: Let f(z) be analytic and univalent in the unit disc \(\Delta:z:<1\). If f and each successive derivative \(f^{(k)}\) are univalent in \(\Delta\), then S. M. Shah and S. Y. Trimble showed that f must be an entire function of exponential type (1969; see also A. Sathaye and S. M. Shah, 1982). This hypothesis that each \(f^{(k)}\) univalent was improved to the condition that each \(f^{(n_ k)}\) \((k=1,2,...)\), where \(\{n_ k\}\) is an increasing sequence of positive integers such that \[ (1)\quad n_{k+2}-2n_{k+1}+n_ k=o(n_ k) \] is univalent in \(\Delta\) '' implies that f is entire. Let \(\{d_ n\}\) be a nondecreasing sequence of positive numbers. The Gelfond-Leontev derivative of \(f(z)=\sum^{\infty}_{0}a_ nz^ n\), \(| z| <R\) is defined by \(Df(z)=\sum^{\infty}_{n=1}d_ na_ nz^{n-1}\) and the kth iterative in D is given by \(D^ kf(z)=\sum^{\infty}_{n=k}d_ n...d_{n-k+1}a_ nz^{n-k}\). O. P. Juneja and S. M. Shah (1984) considered the Gelfond-Leontev derivative for univalence and growth properties, and extended some results of Shah and Trimble. In the present paper the authors prove three theorems. In Theorem 1, f(z) is assumed to be entire and the condition analogues to (1) is assumed for univalence of \(D^{np}f\) and a bound on the growth of f is obtained. In Theorem 2, log d(n) is assumed to be the restriction of a slowly oscillating function on integers.
Mathematics Subject Classification ID: 30C50
Mathematics Subject Classification ID: 30D45
zbMATH DE Number: 4085989
zbMATH Keywords: entire function of exponential type
zbMATH Keywords: Gelfond-Leontev derivative
zbMATH Keywords: slowly oscillating function
MaRDI profile type: Publication
OpenAlex ID: W2100816958
description: scientific article; zbMATH DE number 4085989