  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.
