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∈????, a Banach algebra, with ‖x‖,‖y‖<R we show among others that‖f(y)-f(x)‖ ≤ ‖y-x‖∫01f\u27a(‖(1-t)x+ty‖)dtwhere fa(z) = Σn=0∞|αn|zn. Inequalities for the commutator such as‖f(x)f(y) - f(y)f(x)‖ ≤ 2fa(M)f\u27a(M)‖y-x‖,if ‖x‖,‖y‖≤M<R, as well as some inequalities of Hermite–Hadamard type are also provided
