Inequalities Of Lipschitz Type For Power Series In Banach Algebras

• Volume: 29, Issue: 1, page 61-83
• ISSN: 0860-2107

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Abstract

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Let [...] f(z)=∑n=0∞αnzn $f\left(z\right)={\sum }_{n=0}^{\infty }{\alpha }_{n}{z}^{n}$ 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 $∥f\left(y\right)-f\left(x\right)∥\le ∥y-x∥{\int }_{0}^{1}{f}_{a}^{\text{'}}\left(∥\left(1-t\right)x+ty∥\right)dt$ where [...] fa(z)=∑n=0∞|αn| zn ${f}_{a}\left(z\right)={\sum }_{n=0}^{\infty }|{\alpha }_{n}|\phantom{\rule{0.277778em}{0ex}}{z}^{n}$ . Inequalities for the commutator such as [...] ‖f(x)f(y)−f(y)f(x)‖≤2fa(M)fa′(M)‖y−x‖, $∥f\left(x\right)f\left(y\right)-f\left(y\right)f\left(x\right)∥\le 2{f}_{a}\left(M\right){f}_{a}^{\text{'}}\left(M\right)∥y-x∥,$ if ‖x‖, ‖y‖ ≤ M < R, as well as some inequalities of Hermite–Hadamard type are also provided.

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