Inequalities for Power Series in Banach Algebras

Abstract

For any $x$,$y\in ℬ$, a unital Banach algebra and $n\ge 1$ we show that

$$\Vert y^n-x^n\Vert \le n\Vert y-x\Vert \underset{0}{\overset{1}{{\displaystyle \int }}}\Vert (1-t)x+ty{\Vert }^{n-1}dt.$$

Upper bounds for quantities such as

$$\Vert f(x)-f(y)\Vert ,\Vert f(xy)-f(yx)\Vert ,$$

and

$$\Vert \frac{f(x)+f(y)}{2}-f(\frac{x+y}{2})\Vert$$

that can naturally be associated with the analytic function $f(\lambda ):=\sum _{j=0}^{\infty }{\alpha }_j{\lambda }^j$ defined on the open disk $D(0,R)$ and the elements $x$ and $y$ of the unital Banach algebra $ℬ$ are given. Some applications for functions of interest such as the exponential map on $ℬ$ and the resolvent function are provided as well.