New Norm Inequalities of Čebyšev Type for Power Series in Banach Algebras

Abstract

Let $f\left( \lambda \right) =\sum_{n=0}^{\infty }\alpha _{n}\lambda ^{n}$ be a function defined by power series with complex coefficients and convergent on the open disk $D\left( 0,R\right) \subset \mathbb{C}$, $R>0$ and $x,y\in \mathcal{B}$, a Banach algebra, with $xy=yx.$

In this paper we establish some new upper bounds for the norm of the Čebyšev type difference
\begin{equation*}
f\left( \lambda \right) f\left( \lambda xy\right) -f\left( \lambda x\right)
f\left( \lambda y\right)
\end{equation*}
provide that the complex number $\lambda $ and the vectors $x,y\in \mathcal{B%}$ are such that the series in the above expression are convergent. These results complement the earlier resuls obtained by the authors. Applications
for some fundamental functions such as the exponential function and the resolvent function are provided as well.