New classification of analytic functions with negative coefficients
type: Item
instance of: scholarly article
title: New classification of analytic functions with negative coefficients
zbMATH Open document ID: 0634.30016
DOI: 10.1155/S0161171288000109
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/46243
review text: Let \(A_{p,n}\) denote the class of functions of the form \[ f(z)=z\quad p-\sum^{\infty}_{k=p+n}a_ kz\quad k,\quad a_ k\geq 0, \] p, \(n\in \{1,2,3,...\}\), which are analytic in the unit disc \textit{U. H. Silvermann} [Proc. Am. Math. Soc. 51, 109-116 (1975; Zbl 0311.30007)] has obtained necessary and sufficient conditions in terms of \(\{a_ n\}\) for the class of \(A_{1,n}\) and many authors have obtained such conditions for subclasses of univalent or p-valent analytic functions with negative coefficients. In this paper, the authors unify all the results by defining the class \(A(p,n,B_ k)\) of all functions of \(A_{p,n}\), which satisfy the inequality \(\sum^{\infty}_{k=p+n}B_ ka_ k\leq 1\). Various distortion theorems for these functions and for fractional calculus of functions belonging to \(A(p,n,B_ k)\) are given and some consequences are discussed.
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 26A24
zbMATH DE Number: 4032032
zbMATH Keywords: fractional derivative
zbMATH Keywords: p-valent analytic functions
zbMATH Keywords: distortion theorems
zbMATH Keywords: fractional calculus
author: Shigeyoshi Owa
author: M. Obradović
reviewed by: O. Fekete
MaRDI profile type: Publication
OpenAlex ID: W2165499291
description: scientific article; zbMATH DE number 4032032