Levin Steckin theorem and inequalities of the Hermite-Hadamard type

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Recently Ohlin lemma on convex stochastic ordering was used to obtain some inequalities of Hermite-Hadamard type. Continuing this idea, we use Levin-Stečkin result to determine all inequalities of the forms: ∑ᵢ₌₁³aᵢf(αᵢx+(1-αᵢ)y)≤ 1/y-x∫ₓʸf(t), a₁f(x)+∑ᵢ₌₂³aᵢf(αᵢx+(1-αᵢ)y)+a₄f(y)≥ 1/y-x∫ₓʸf(t) and af(α₁x+(1-α₁)y)+(1-a)f(α₂x+(1-α₂)y)≤ b₁f(x)+b₂f(βx+(1-β)y)+b₃f(y) which are satisfied by all convex functions f:[x,y]→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.