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Annales Mathematicae Silesianae

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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
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Published Online: 2015-09-30 | DOI: https://doi.org/10.1515/amsil-2015-0006

Abstract

Let f(z)=n=0αnzn 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)yx01fa((1t)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)yx, 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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About the article

Received: 2014-10-21

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, DOI: https://doi.org/10.1515/amsil-2015-0006.

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© Sever S. Dragomir. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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