Ohlin’s lemma and some inequalities of the Hermite–Hadamard type

Abstract

Abstract. 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] such that for all convex functions f the inequalityis satisfied and all a, b, c, α ∈ (0, 1) with a + b + c = 1 for which we haveMathematics Subject Classification. Primary 26A51; Secondary 26D10, 39B62.

A solution to the problem of Raşa connected with Bernstein polynomials

“…Therefore a simple application of Ohlin's Lemma is impossible and an extra idea is needed. To handle such situations, in the papers [9,12], the authors employed the Levin-Stečkin theorem [6] (see also [7], Theorem 4.2.7). for all continuous convex functions f : [a, b] → R, it is necessary and sufficient that F 1 and F 2 satisfy the following three conditions: To start our considerations, we define the number of sign changes of a function ϕ : R → R by…”

“…Remark 1. Szostok noticed in [12] that if the measures µ X , µ Y corresponding to X , Y , respectively, are concentrated on the interval [a, b], then, in fact, the relation X cx Y holds if and only if the inequality (1.2) is satisfied for all continuous convex functions f : [a, b] → R.…”

On Some Recent Applications of Stochastic Convex Ordering Theorems to Some Functional Inequalities for Convex Functions: A Survey

“…In the paper [40], the author used Ohlin's lemma to prove some new inequalities of the Hermite-Hadamard type, which are a generalization of known Hermite-Hadamard type inequalities.…”

“…This means that (42) provides an alternate proof of (41) (for twice differentiable f ). This new formulation of the Hermite-Hadamard inequality was inspiration in [32] to replace the middle term of Hermite-Hadamard inequality by more complicated expressions than those used in (40). In [32], the authors study inequalities of the form…”

Functional Inequalities Involving Numerical Differentiation Formulas of Order Two

“…we can see that (2) is, in fact, an inequality involving two very simple quadrature operators and a very simple differentiation formula. In papers [11] and [12] the quadrature operators occurring in (2) were replaced by more general ones whereas in [9] the middle term from (2) was replaced by more general formulas used in numerical differentiation. Thus inequalities involving expressions of the form n i=1 a i F (α i x + β i y) y − x where n i=1 a i = 0, α i + β i = 1 and F ′ = f were considered.…”