Levin Steckin theorem and inequalities of the Hermite-Hadamard type
  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. 
Tomasz Szostok
math.CA