Levin Steckin theorem and inequalities of the Hermite-Hadamard type

Abstract

Recently Ohlin lemma on convex stochastic ordering was used to obtain some inequalities of Hermite-Hadamard type. Continuing this idea, we use Levin-Stečkin result to determine all inequalities of the forms: $$\sum _{i=1}^3{a}_{i}f({\alpha }_{i}x+(1-{\alpha }_i)y)\le \frac{1}{y-x}{\int }_x^{y}f(t),$$ $$a_{1}f(x)+\sum _{i=2}^3{a}_{i}f({\alpha }_{i}x+(1-{\alpha }_i)y)+a_{4}f(y)\ge \frac{1}{y-x}{\int }_x^{y}f(t)$$ and $$af({\alpha }_{1}x+(1-{\alpha }_1)y)+(1-a)f({\alpha }_{2}x+(1-{\alpha }_2)y)\le b_{1}f(x)+b_{2}f(\beta x+(1-\beta )y)+b_{3}f(y)$$ which are satisfied by all convex functions $f:[x,y]\to \mathbb{R}.$ As it is easy to see, the same methods may be applied to deal with longer expressions of the forms considered. As particular cases of our results we obtain some known inequalities.

References in corpus (1)

• Inequalities of the Hermite-Hadamard type involving numerical differentiation formulas