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# Annales Mathematicae Silesianae

Editor-in-Chief: Sablik, Maciej

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Mathematical Citation Quotient (MCQ) 2016: 0.10

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Volume 29, Issue 1

# Inequalities Of Lipschitz Type For Power Series In Banach Algebras

Sever S. Dragomir
• Mathematics, School of Engineering & Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia
• School of Computer Science & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa, URL: http://rgmia.org/dragomir
• Email
• Other articles by this author:
Published Online: 2015-09-30 | DOI: https://doi.org/10.1515/amsil-2015-0006

## Abstract

Let $f(z)=∑n=0∞αnzn$ be a function defined by power series with complex coefficients and convergent on the open disk D (0, R) ⊂ ℂ, R > 0. For any x, y ∈ ℬ, a Banach algebra, with ‖x‖, ‖y‖ < R we show among others that $‖f(y)−f(x)‖≤‖y−x‖∫01fa′(‖(1−t)x+ty‖)dt$ where $fa(z)=∑n=0∞|αn| zn$ . Inequalities for the commutator such as $‖f(x)f(y)−f(y)f(x)‖≤2fa(M)fa′(M)‖y−x‖,$ if ‖x‖, ‖y‖ ≤ M < R, as well as some inequalities of Hermite–Hadamard type are also provided.

(2010) Mathematics Subject Classification:: 47A63; 47A99

Key words and phrases:: Banach algebras; Power series; Lipschitz type inequalities; Hermite-Hadamard type inequalities

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Revised: 2015-03-04

Published Online: 2015-09-30

Published in Print: 2015-09-01

Citation Information: Annales Mathematicae Silesianae, Volume 29, Issue 1, Pages 61–83, ISSN (Online) 0860-2107,

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