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: $$\sum _{i=1}^3{a}_{i}f({\alpha }_{i}x+(1-{\alpha }_i)y)\le \frac{1}{y-x}{\int }_x^{y}f(t),$$ $$a_{1}f(x)+\sum _{i=2}^3{a}_{i}f({\alpha }_{i}x+(1-{\alpha }_i)y)+a_{4}f(y)\ge \frac{1}{y-x}{\int }_x^{y}f(t)$$ 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 $f:[x,y]\to \mathbb{R}\mathrm{.}$ 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.