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-Ste\v{c}kin result to determine all inequalities of the forms: $$\sum_{i=1}^3a_if(\alpha_ix+(1-\alpha_i)y)\leq \frac{1}{y-x}\int_{x}^yf(t),$$ $$a_1f(x)+\sum_{i=2}^3a_if(\alpha_ix+(1-\alpha_i)y)+a_4f(y)\geq \frac{1}{y-x}\int_{x}^yf(t)$$ and $$af(\alpha_1x+(1-\alpha_1)y)+(1-a)f(\alpha_2x+(1-\alpha_2)y)\leq b_1f(x)+b_2f(\beta x+(1-\beta)y)+b_3f(y)$$ 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.