Two points and $n$ th derivatives norm inequalities for analytic functions in Banach algebras

S. S. Dragomir

doi:10.37863/umzh.v74i8.6116

S. S. Dragomir College Engineering and Sci., Victoria Univ., Melbourne, Australia; School Comput. Sci. and Appl.Math., Univ. Witwatersrand, Johannesburg, South Africa

DOI: https://doi.org/10.37863/umzh.v74i8.6116

Keywords: Banach algebras, Ostrowski inequality, Norm inequalities, Analytic functional calculus.

Abstract

UDC 517.5

Let $\mathcal{B}$ be a unital Banach algebra, let $a \in \mathcal{B},$ $G$ be a convex domain of $\mathbb{C}$ with $\sigma (a) \subset G,$ let $\alpha, \beta \in G,$ and let $f \colon G \rightarrow \mathbb{C}$ be analytic on $G.$
By using the analytic functional calculus, we obtain among others the following result:
\begin{gather*}
\left\| f(a) - \frac{1}{2}\sum_{k=0}^{n}\frac{1}{k!}\left[
f^{(k) }(\alpha) (a-\alpha)
^{k}+(-1)^{k}f^{(k) }(\beta) (\beta-a)^{k}\right] \right\|\leq
\\
\leq \frac{1}{2(n+1) !}\left[\|a-\alpha\|
^{n+1}+\|\beta - a\|^{n+1}\right]\times
\\
\times \max \left\{\sup_{s\in [0,1] }\left\| f^{(n+1) }[(1-s) \alpha+sa] \right\|,\sup_{s\in [0,1] }\left\| f^{(n+1) }[(1-s) a+s\beta] \right\| \right\}.
\end{gather*}
Some examples for the exponential function on Banach algebras are also given.

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