Abstract
Let $\mathcal{B}$ be a unital Banach algebra, $a\in \mathcal{B}$, $G$ be a convex domain of $\mathbb{C}$ with $\sigma \left( a\right) \subset G$ and $\gamma \subset G$ is a piecewise smooth path parametrized by $\lambda\left( t\right)$, $t\in \left[ 0,1\right]$ from $\lambda \left( 0\right)=\alpha$ to $\lambda \left( 1\right) =\beta$, with $\beta \neq \alpha$. If $f:G\rightarrow \mathbb{C}$ is analytic on $G$, then by using the analytic functional calculus we obtain among others the following result $\begin{multline*} \left\Vert f\left( a\right) -\int_{0}^{1}f\left( \left( 1-t\right) \lambda +ta\right) dt\right\Vert \leq \left\Vert a-\lambda \right\Vert \int_{0}^{1}t\left\Vert f^{\prime }\left( ta+\left( 1-t\right) \lambda \right) \right\Vert dt \\ \leq \left\Vert a-\lambda \right\Vert \left\{ \begin{array}{l} \frac{1}{2}\sup_{t\in \left[ 0,1\right] }\left\Vert f^{\prime }\left( ta+\left( 1-t\right) \lambda \right) \right\Vert , \\ \\ \frac{1}{\left( q+1\right) ^{1/q}}\left( \int_{0}^{1}\left\Vert f^{\prime }\left( ta+\left( 1-t\right) \lambda \right) \right\Vert ^{p}dt\right) ^{1/p}, \\ \\ \int_{0}^{1}\left\Vert f^{\prime }\left( ta+\left( 1-t\right) \lambda \right) \right\Vert dt,% \end{array} \right. \end{multline*}$ for all $\lambda \in G$. Some example for the exponential function of elements in Banach algebras are also provided.
Citation
Silvestru Sever Dragomir. "Some Norm Integral Inequalities for Analytic Functions in Banach Algebras." Commun. Math. Anal. 23 (1) 63 - 81, 2020.
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