On certain classes of \(p\)-valent analytic functions
type: Item
instance of: scholarly article
title: On certain classes of \(p\)-valent analytic functions
zbMATH Open document ID: 0773.30008
DOI: 10.1155/S0161171293000389
author: Nak Eun Cho
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1993-10-21T00:00:00Z
full work available at URL: https://eudml.org/doc/46859
review text: The author introduces the class \(P(p,\alpha)\) \((0\leq \alpha < p)\) \((p=1,2,\dots)\) consisting of functions \(f(z)\) of the form (1) \(f(z)=z^ p+\sum^ \infty_{n=1} a_{n+p} z^{n+p}\cdots\) which are analytic and satisfying \(\text{Re}\bigl[ f'(z)/z^{p-1}\bigr]>\alpha\), in the open unit disc of the complex plane. On the other hand \(f(z)\) of the form (1) is said to belong to \(P(p,\alpha,\beta)\) if \((p+1)^{-\beta} D^ \beta f\in P(p,\alpha)\) \((\beta\geq 0)\). In this paper the author obtains some inclusion theorems for the various classes \(P(p,\alpha,\beta)\) and the effect of certain integral operators on these families. For example it is proved that if \(f\in P(p,\alpha,\beta)\) then \((1+p)^ \delta I^ \delta f(z)\) also belongs to \(P(p,\alpha,\beta)\), where \[ I^ \delta f(z)={1\over \Gamma(z)} \int^ 1_ 0 \bigl(\log(1/t)\bigr)^{\delta-1} f(zt)dt\quad (\delta>0). \] {}.
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 179251
zbMATH Keywords: \(p\)-valent analytic functions
reviewed by: V. Karunakaran
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 179251