SOME BOUNDS FOR THE COMPLEX µCEBYŠEV FUNCTIONAL OF ABSOLUTELY CONTINUOUS FUNCTIONS

Abstract

In this paper we provide several bounds for the modulus of the \textit{%complex \v{C}eby\v{s}ev functional}%\begin{equation*}C\left( f,g\right) :=\frac{1}{b-a}\int_{a}^{b}f\left( t\right) g\left(t\right) dt-\frac{1}{b-a}\int_{a}^{b}f\left( t\right) dt\int_{a}^{b}g\left(t\right) dt\end{equation*}%under various assumptions for the integrable functions $f,$ $g:\left[ a,b%\right] \rightarrow \mathbb{C}$. We show amongst others that, if $f$ and $g$are absolutely continuous on $\left[ a,b\right] $ with $f^{\prime }\in L_{p}%\left[ a,b\right] ,$ $g^{\prime }\in L_{q}\left[ a,b\right] ,$ $p,$ $q>1$and $\frac{1}{p}+\frac{1}{q}=1$, then%\begin{equation*}\max \left\{ \left\vert C\left( f,g\right) \right\vert ,\left\vert C\left(\left\vert f\right\vert ,g\right) \right\vert ,\left\vert C\left(f,\left\vert g\right\vert \right) \right\vert ,\left\vert C\left( \left\vertf\right\vert ,\left\vert g\right\vert \right) \right\vert \right\}\end{equation*}%\begin{equation*}\leq \left[ C\left( \ell ,F_{\left\vert f^{\prime }\right\vert ^{p}}\right) %\right] ^{1/p}\left[ C\left( \ell ,F_{\left\vert g^{\prime }\right\vert^{q}}\right) \right] ^{1/q},\end{equation*}%where $F_{\left\vert h\right\vert }:\left[ a,b\right] \rightarrow \mathbb{[}%0,\infty )$ is defined by $F_{\left\vert h\right\vert }\left( t\right):=\int_{a}^{t}$.$\left\vert h\left( t\right) \right\vert dt$ and $\ell :%\left[ a,b\right] \rightarrow \left[ a,b\right] ,$ $\ell \left( t\right) =t$is the identity function on the interval $\left[ a,b\right] .$ Applicationsfor the trapezoid inequality are also provided.