Certain integral operators on the classes \(\mathcal M(\beta_i)\) and \(\mathcal N(\beta_i)\)
type: Item
instance of: scholarly article
title: Certain integral operators on the classes \(\mathcal M(\beta_i)\) and \(\mathcal N(\beta_i)\)
zbMATH Open document ID: 1149.30014
DOI: 10.1155/2008/719354
author: Daniel Breaz
published in: Journal of Inequalities and Applications
publication date: 2008-08-19T00:00:00Z
review text: The author considers the classes \(M(\beta)\) and \(N(\beta)\) of analytic functions and two general integral operators. He proves some properties for these operators on these classes. The class \(M(\beta)\) is the subclass of \(A\), consisting of the functions \(f(z)\), which satisfy the inequality \[ \text{Re}\left \{ \frac{zf'(z)}{f(z)}\right\}< \beta,\quad z\in U,\;\beta >1, \] and \(N(\beta)\) is the subclass of \(A\), consisting of the functions \(f(z)\), which satisfy the inequality \[ \text{Re}\left \{ \frac{zf''(z)}{f'(z)}+1\right \}< \beta,\quad z\in U. \] The integral operators studied in this paper are \[ F_{n}(z)=\int_{0}^{z} \left( \frac{f_{1}(t)}{t}\right)^{\alpha_{1}}\cdots \left( \frac{f_{n}(t)}{t}\right)^{\alpha_{n}}\,dt, \] where \(f_{i}(z)\in A\) and \(\alpha_{i} >0\), for all \(i\in \{1,2,\dots ,n\}\), and \[ F_{\alpha_{1},\dots ,\alpha_{n}}(z)= \int_{0}^{z} [f_{1}'(t)]^{\alpha_{1}}\cdots [f_{n}^{\prime}(t)]^{\alpha_{n}} \,dt, \] where \(f_{i}(z)\in A\) and \(\alpha_{i} >0\), for all \(i\in \{1,2,\dots,n\}\).
reviewed by: Dorin Blezu
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 5313199
zbMATH Keywords: subclasses of analytic functions
Wikidata QID: Q59216481
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2008/719354
OpenAlex ID: W2083034982
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description: scientific article; zbMATH DE number 5313199