Inequalities Of Lipschitz Type For Power Series In Banach Algebras

Sever S. Dragomir

Annales Mathematicae Silesianae, Tome 29 (2015), p. 61-83 / Harvested from The Polish Digital Mathematics Library

Résumé

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 f(y)-f(x)≤y-x∫01fa'((1-t)x+ty)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‖, 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.

Détail

Publié le : 2015-01-01
EUDML-ID : urn:eudml:doc:276889

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