Let
$f(z) = \sum\nolimits_{n = 0}^\infty {\alpha _n z^n }$
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
$$\left\| {f(y) - f(x)} \right\| \le \left\| {y - x} \right\|\int_0^1 {f_a^\prime } (\left\| {(1 - t)x + ty} \right\|)dt$$
where
$f_a (z) = \sum\nolimits_{n = 0}^\infty {|\alpha _n |} \;z^n$
. Inequalities for the commutator such as
$$\left\| {f(x)f(y) - f(y)f(x)} \right\| \le 2f_a (M)f_a^\prime (M)\left\| {y - x} \right\|,$$
if ‖x‖, ‖y‖ ≤ M < R, as well as some inequalities of Hermite–Hadamard type are also provided.