Using the Ohlin lemma on convex stochastic ordering we prove inequalities of the Hermite–Hadamard type. Namely, we determine all numbers a,α,β∈[0,1]\documentclass[12pt]{minimal}
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\begin{document}$${a,\alpha,\beta\in[0,1]}$$\end{document} such that for all convex functions f the inequality
af(αx+(1-α)y)+(1-a)f(βx+(1-β)y)≤1y-x∫xyf(t)dt\documentclass[12pt]{minimal}
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\begin{document}$$af(\alpha x+(1-\alpha )y)+(1-a)f(\beta x+(1-\beta) y)\leq \frac{1}{y-x} \int\limits_{x}^yf(t)dt$$\end{document}is satisfied and all a,b,c,α∈(0,1)\documentclass[12pt]{minimal}
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\begin{document}$${a,b,c,\alpha\in(0,1)}$$\end{document} with a + b + c = 1 for which we have
af(x)+bf(αx+(1-α)y)+cf(y)≥1y-x∫xyf(t)dt\documentclass[12pt]{minimal}
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\begin{document}$$af(x)+bf(\alpha x+(1-\alpha)y)+cf(y)\geq\frac{1}{y-x} \int\limits_{x}^yf(t)dt$$\end{document}.
