Some Norm Integral Inequalities for Analytic Functions in Banach Algebras

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.