Inequalities related to the Chebyshev functional involving integrals over different intervals (Q2740877)

The generalized Chebyshev functional is defined by NEWLINE\[NEWLINET(f,g;a,b,c,d)=M(fg; a,b)+ M(fg;c,d)- M(f; a,b)M(g; c,d)- M(f;c,d) M(g;a,b),NEWLINE\]NEWLINE where \(M(f;a,b)\) is the integral mean of \(f\) on \([a,b]\). Let us suppose that \(f,g:I\to \mathbb{R}\) are measurable on \(I\) and \([a,b], [c,d]\subset I\). Further suppose that \(|f(x)- f(y)|\leq H_1|x-y|^r\) and \(|g(x)- g(y)|\leq H_2|x-y|^s\) for all \(x\in [a,b]\), \(y\in [c,d]\) and certain \(H_1,H_2> 0\); \(r,s\in (0,1]\). If the integrals involved exist, then one has NEWLINE\[NEWLINE\begin{multlined} (k+1)(k+2)|T(f,g; a,b,c,d)|\\ \leq((H_1H_2)/(b- a)(d- c))[|b-c|^{k+2}-|b-d|^{k+2}+|d-a|^{k+2}-|c-a|^{k+2}],\end{multlined}NEWLINE\]NEWLINE where \(k= r+s\). The Lipschitzian case is also considered. Other results involve a weighted generalized Chebyshev functional.