Integral Inequalities

Abstract

1. To G. Griiss (1934) [l] is due the inequality 1):

$$|\frac{1}{{{\text{b - a}}}}\int\limits_{\text{a}}^{\text{b}} {\text{f}} \left( {\text{x}} \right){\text{g}}\left( {\text{x}} \right){\text{dx|}} \leqslant \frac{1}{4}\mathop {{\text{Osc}}\,{\text{f}}\,{\text{Osc}}\,{\text{g}}}\limits_{<{\text{a,b}}> \,\,\, <{\text{a,b}}>} \left( {\int\limits_{\text{a}}^{\text{b}} {{\text{f}}\left( {\text{x}} \right)} \,{\text{dx}}\,{\text{ = }}\,{\text{0}}} \right)$$

(1)

Here the condition $\int_{\text{a}}^{\text{b}} {\text{f}} \left( {\text{x}} \right){\text{dx}}\,{\text{ = }}\,{\text{0}}$ can be dropped, if we replace the integral mean on the left with the expression

$${\text{T }}\left( {{\text{f, g}}} \right) = \frac{1}{{{\text{b - a}}}}\int\limits_{\text{a}}^{\text{b}} {\text{f}} \left( {\text{x}} \right){\text{g}}\left( {\text{x}} \right){\text{dx - }}\frac{1}{{{\text{b - a}}}}\int\limits_{\text{a}}^{\text{b}} {{\text{f}}\left( {\text{x}} \right){\text{dx}} \cdot \frac{1}{{{\text{b - a}}}}\int\limits_{\text{a}}^{\text{b}} {{\text{g}}\left( {\text{x}} \right){\text{dx}}} } $$

(2)