Inequalities of Lipschitz type for power series in Banach algebras
Abstract
Let f(z) = Σn=0∞αnzn 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‖∫01f'a(‖(1-t)x+ty‖)dt
where fa(z) = Σn=0∞|αn|zn. Inequalities for the commutator such as
‖f(x)f(y) - f(y)f(x)‖ ≤ 2fa(M)f'a(M)‖y-x‖,
if ‖x‖,‖y‖≤M<R, as well as some inequalities of Hermite–Hadamard type are also provided.
Keywords
Banach algebras; Power series; Lipschitz type inequalities; Hermite-Hadamard type inequalities
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Mathematics, School of Engineering & Science, Victoria University, Australia Australia
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