On trapezoid inequality via a Grüss type result and applications (Q2724875)

The following result of Grüss type is proved:NEWLINENEWLINENEWLINELet \(f,g:[a,b]\to \mathbb{R}\) be two integrable mappings. Then NEWLINE\[NEWLINE\begin{multlined}\Biggl|{1\over b-a} \int^b_a f(x)g(x) dx- {1\over b-a} \int^b_a f(x) dx\cdot{1\over b-a} \int^b_a g(x) dx\Biggr|\leq\\ {1\over b-a} \int^b_a\Biggl|\Biggl(f(x)- {1\over b-a} \int^b_a f(y) dy\Biggr) \Biggl(g(x)- {1\over b-a} \int^b_a g(y) dy\Biggr)\Biggr|dx.\end{multlined}NEWLINE\]NEWLINE The inequality is sharp.NEWLINENEWLINENEWLINESome applications are also given.