Inequalities Of Lipschitz Type For Power Series In Banach Algebras

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Abstract

Let f(z)=n=0αnzn 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)yx01fa((1t)x+ty)dt
where fa(z)=n=0|αn|zn . Inequalities for the commutator such as
f(x)f(y)f(y)f(x)2fa(M)fa(M)yx,
if ‖x‖, ‖y‖ ≤ M < R, as well as some inequalities of Hermite–Hadamard type are also provided.

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