Levin Steckin theorem and inequalities of the Hermite-Hadamard type

2014/11/27 by Tomasz Szostok, Szostok, Tomasz
Mathematics · #26A51 #26D10 #39B62 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Equations Stability Results #Mathematical Inequalities and Applications #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1411.7708

openalex publication_date 2014/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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=1³a_if(α_ix+(1-α_i)y)≤ (1)/(y-x)∫_x^yf(t), a₁f(x)+∑_i=2³a_if(α_ix+(1-α_i)y)+a₄f(y)≥ (1)/(y-x)∫_x^yf(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]→\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.

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