## Inequalities of Hadamard’s type for Lipschitzian mappings and their applications.(English)Zbl 0956.26015

The authors prove the following two inequalities of Hadamard’s type: Let $$f:I\subseteq \mathbb{R\rightarrow R}$$ be an $$M$$-Lipschitzian mapping on $$I$$ and $$a,b\in I$$ with $$a < b$$. Then we have the inequalities $\left|f\left( \frac{a+b}{2}\right) -\frac{1}{b-a}\int_{a}^{b}f(x) dx\right|\leq \frac{M}{4}(b-a)$ and $\left|\frac{f(a)+f(b)}{2}- \frac{1} {b-a} \int_{a}^{b}f(x) dx\right|\leq \frac{M}{3}(b-a).$ Some applications for means of two positive numbers are also given.

### MSC:

 26D15 Inequalities for sums, series and integrals 26E60 Means 26A16 Lipschitz (Hölder) classes

### Keywords:

Hadamard’s inequality; $$M$$-Lipschitzian mapping; means
Full Text: