A Grüss' type integral inequality for mappings of \(r\)-Hölder's type and applications for trapezoid formula (Q2719885)

Let the real numbers \(r\in (0,1]\) and \(H>0\) be fixed. The function \(f\) is called of \(r-H\)-Hölder type on \([a,b]\) if NEWLINE\[NEWLINE \left|f(x)-f(y)\right|\leq H\cdot \left|x-y\right|^{r},\quad\forall x,y\in \left[ a,b\right] . NEWLINE\]NEWLINE The main result of this paper is the following Grüss' type inequality: if \(f\) is of \(r-H_{1}\)-Hölder type and \(g\) is of \(s-H_{2}\)-Hölder type on \([a,b]\), then we have the inequality NEWLINE\[NEWLINE \left|\frac{1}{b-a}\int_{a}^{b}f(x)g(x) dx-\frac{1}{b-a}\int_{a}^{b}f(x) dx \cdot \frac{1}{b-a}\int_{a}^{b}g(x)dx\right|\leq \frac{H_{1}H_{2}(b-a)^{r+s} }{(r+s+1)(r+s+2)}. NEWLINE\]NEWLINE Some applications for the trapezoid formula in numerical integration are given.