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Inequalities Of Lipschitz Type For Power Series In Banach Algebras

Sever S. Dragomir — 2015

Annales Mathematicae Silesianae

Let [...] f(z)=∑n=0∞αnzn f ( z ) = ∑ n = 0 ∞ α 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 ∫ 0 1 f a ' ( ( 1 - t ) x + t y ) d t where [...] fa(z)=∑n=0∞|αn| zn f a ( z ) = ∑ n = 0 ∞ | α 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 ) ≤ 2 f a ( M ) f a ' ( M ) y - x , if ‖x‖, ‖y‖ ≤ M < R, as well as some inequalities of Hermite–Hadamard type are also provided.

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