Inequalities Of Lipschitz Type For Power Series In Banach Algebras

Abstract

Abstract Let f ( z ) = ∑ n = 0 ∞ α n z n $f(z) = \sum\nolimits_{n = 0}^\infty {\alpha _n z^n }$ 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 ‖ ∫ 0 1 f a ′ ( ‖ ( 1 − t ) x + t y ‖ ) d t $$\left\| {f(y) - f(x)} \right\| \le \left\| {y - x} \right\|\int_0^1 {f_a^\prime } (\left\| {(1 - t)x + ty} \right\|)dt$$ where f a ( z ) = ∑ n = 0 ∞ | α n | z n $f_a (z) = \sum\nolimits_{n = 0}^\infty {|\alpha _n |} \;z^n$ . Inequalities for the commutator such as ‖ f ( x ) f ( y ) − f ( y ) f ( x ) ‖ ≤ 2 f a ( M ) f a ′ ( 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.