Inequalities of Lipschitz type for power series in Banach algebras

Sever S. Dragomir

Published: 2015-09-30

Vol. 29 (2015)

Inequalities of Lipschitz type for power series in Banach algebras

Sever S. Dragomir

Abstract

Let f(z) = Σ_n=0^∞α_nzⁿ 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∈????, a Banach algebra, with ‖x‖,‖y‖<R we show among others that
‖f(y)-f(x)‖ ≤ ‖y-x‖∫₀¹f'_a(‖(1-t)x+ty‖)dt
where f_a(z) = Σ_n=0^∞|α_n|zⁿ. Inequalities for the commutator such as
‖f(x)f(y) - f(y)f(x)‖ ≤ 2f_a(M)f'_a(M)‖y-x‖,
if ‖x‖,‖y‖≤M<R, as well as some inequalities of Hermite–Hadamard type are also provided.

Keywords:

Banach algebras , Power series , Lipschitz type inequalities , Hermite-Hadamard type inequalities

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Dragomir, S. S. (2015). Inequalities of Lipschitz type for power series in Banach algebras. Annales Mathematicae Silesianae, 29, 61–83. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13978

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Vol. 29 (2015)
Published: 2015-09-30

ISSN: 0860-2107

eISSN: 2391-4238

10.2478/amsil

Publisher

University of Silesia Press

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