×

Integral inequalities of Grüss type via Pólya-Szegö and Shisha-Mond results. (English) Zbl 1047.26013

Let \(f,g:[a,b]\rightarrow \mathbb{R}\) be two integrable functions so that \( m\leq f\leq M\) and \(n\leq g\leq N\) a.e. on \([a,b]\). Define the functional \( A(f;a,b)=\frac{1}{b-a}\int_{a}^{b}f(x)dx\) and then the Chebychev’s functional \(T(f,g;a,b)=A(f\cdot g;a,b)\) \(-A(f;a,b)\cdot A(g;a,b)\). Grüss’ inequality is given by \(\left| T(f,g;a,b)\right| \leq \frac{1}{4} \left( M-m\right) \left( N-n\right) .\) For \(m,n>0,\) the authors prove two new inequalities: \[ \left| T(f,g;a,b)\right| \leq \frac{1}{4}\cdot \frac{\left( M-m\right) \left( N-n\right) }{\sqrt{mnMN}}\cdot A(f;a,b)\cdot A(g;a,b) \] and \[ \left| T(f,g;a,b)\right| \leq \left( \sqrt{M}-\sqrt{m} \right) \left( \sqrt{N}-\sqrt{n}\right) \sqrt{A(f;a,b)\cdot A(g;a,b)}. \] They also show that the three inequalities are not related.

MSC:

26D15 Inequalities for sums, series and integrals
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)