Several Grüss’ type inequalities for the complex integral

Abstract

Assume that f and g are continuous on $\gamma$, $\gamma \subset \mathbb { C}$ is a piecewise smooth path parametrized by $z\left( t\right) ,$ $t\in \left[ a,b\right]$ from $z\left( a\right) =u$ to $z\left( b\right) =w$ with $w\ne u$ and the complex Čebyšev functional is defined by

$$\begin{aligned} \mathcal {D}_{\gamma }\left( f,g\right) :=\frac{1}{w-u}\int _{\gamma }f\left( z\right) g\left( z\right) dz-\frac{1}{w-u}\int _{\gamma }f\left( z\right) dz \frac{1}{w-u}\int _{\gamma }g\left( z\right) dz. \end{aligned}$$

In this paper we establish some bounds for the magnitude of the functional $\mathcal {D}_{\gamma }\left( f,g\right)$ and a related version of this under various assumptions for the functions f and g and provide some examples for circular paths.