Nihonkai Math. J. 33 (1), 1-24, (2022)
KEYWORDS: Banach algebras, integral inequalities, analytic functions, Exponential on Banach algebra, 47A63, 47A99

Let $\mathcal{B}$ be a complex Banach algebra. For two continuous functions $x$, $y:\left[a,b\right]\to \mathcal{B}$ we define the *noncommutative Čebyšev functional*

$$D\left(x,y\right):=\left(b-a\right){\displaystyle {\int}_{a}^{b}x}\left(t\right)y\left(t\right)dt-{\displaystyle {\int}_{a}^{b}x}\left(t\right)dt{\displaystyle {\int}_{a}^{b}y}\left(t\right)dt.$$

In this paper we show among other that if $x$, $y$ are strongly differentiable, then

$$\Vert D\left(x,y\right)\Vert \le \frac{1}{4}{\left(b-a\right)}^{2}\times \{\begin{array}{l}{\Vert {x}^{\prime}\Vert}_{\left[a,b\right],1}{\Vert {y}^{\prime}\Vert}_{\left[a,b\right],1},\hfill \\ \frac{1}{3}{\left(b-a\right)}^{2}{\Vert {x}^{\prime}\Vert}_{\left[a,b\right],\infty}{\Vert {y}^{\prime}\Vert}_{\left[a,b\right],\infty}\hfill \\ \frac{1}{2}\left(b-a\right){\Vert {x}^{\prime}\Vert}_{\left[a,b\right],1}{\Vert {y}^{\prime}\Vert}_{\left[a,b\right],\infty},\hfill \end{array}$$

where

$${\Vert {z}^{\prime}\Vert}_{\left[a,b\right],1}:={\displaystyle {\int}_{a}^{b}\Vert {z}^{\prime}\left(t\right)\Vert}dt\text{and}{\Vert {z}^{\prime}\Vert}_{\left[a,b\right],\infty}:=\underset{t\in \left(a,b\right)}{\mathrm{sup}}\Vert {z}^{\prime}\left(t\right)\Vert $$

for a strongly differentiable function $z$ on $\left(a,b\right)$. Some applications for analytic functions of elements in Banach algebras with examples for exponential function are also given.