New Norm Inequalities of \V Ceby\V Sev Type for Power Series in Banach Algebras

S. S. Dragomir, M. V. Boldea, M. Megan

Let $f(\lambda)=\sum_{n=0}^{\infty }\alpha_{n}\lambda^{n}$ be a function defined by power series with complex coefficients and convergent on the open disk $D(0,R)\subset\Bbb{C}$, $R>0$ and $x,y\in\cal{B}$, a Banach algebra, with $xy=yx$. In this paper we establish some new upper bounds for the norm of the \emph{Čebyšev type difference} \[ f(ambda)f(ambda xy)-f(ambda x)f(ambda y), \] providing that the complex number $\lambda$ and the vectors $x,y\in\cal{B}$ 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 \emph{exponential function} and the \emph{resolvent function} are provided as well.