Inequalities Of Lipschitz Type For Power Series In Banach Algebras

• Autorzy: Sever S. Dragomir
• Języki publikacji: EN

Abstrakty

EN

Let [...] f(z)=∑n=0∞αnzn $f(z) = \sum\nolimits_{n = 0}^\infty {\alpha _n z^n }$ 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 $$\left\| {f(y) - f(x)} \right\| \le \left\| {y - x} \right\|\int_0^1 {f_a^\prime } (\left\| {(1 - t)x + ty} \right\|)dt$$ where [...] fa(z)=∑n=0∞|αn| zn $f_a (z) = \sum\nolimits_{n = 0}^\infty {|\alpha _n |} \;z^n$ . Inequalities for the commutator such as [...] ‖f(x)f(y)−f(y)f(x)‖≤2fa(M)fa′(M)‖y−x‖, $$\left\| {f(x)f(y) - f(y)f(x)} \right\| \le 2f_a (M)f_a^\prime (M)\left\| {y - x} \right\|,$$ if ‖x‖, ‖y‖ ≤ M < R, as well as some inequalities of Hermite–Hadamard type are also provided.

Słowa kluczowe

EN

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

Kategorie tematyczne

• 47A63: Operator inequalities
• 47A99: None of the above, but in this section

• Wydawca: De Gruyter Open
• Czasopismo: Annales Mathematicae Silesianae
• Rocznik: 2015
• Tom: 29
• Numer: 1
• Strony: 61-83

Daty

wydano

2015-09-01

otrzymano

2014-10-21

poprawiono

2015-03-04

online

2015-09-30

Twórcy

autor

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:

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Identyfikatory

DOI

10.1515/amsil-2015-0006