Functional Inequalities Involving Numerical Differentiation Formulas of Order Two

Abstract

We write expressions connected with numerical differentiation formulas of order 2 in the form of Stieltjes integral, then we use Ohlin lemma and LevinStechkin theorem to study inequalities connected with these expressions. In particular, we present a new proof of the inequalitysatisfied by every convex function f : R → R and we obtain extensions of this inequality. Then we deal with non-symmetric inequalities of a similar form.

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

“…which we use to approximate the second order derivative of F and, surprisingly, we discover a connection between our approach and the inequality (55) (see [42]). First, we make the following simple observation.…”

“…for some ξ ∈ (x − h, x + h). 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 the paper [42], expressions connected with numerical differentiation formulas of order 2, are studied. The author used the Ohlin lemma and the Levin-Stečkin theorem to study inequalities connected with these expressions.…”

“…In the paper [10], the Hermite-Hadamard type inequalities are interpreted in terms of the convex stochastic ordering between random variables. Recently, also in [19,32,35,36,37,38,40,41,42]), the Hermite-Hadamard inequalities are studied based on the convex ordering properties. Here we want to attract the readers attention to some selected topics by presenting some theorems on the convex ordering that can be useful in the study of the Hermite-Hadamard type inequalities.…”

“…The Ohlin lemma [31] on sufficient conditions for convex stochastic ordering was first used in [36], to get a simple proof of some known Hermite-Hadamard type inequalities as well as to obtaining new Hermite-Hadamard type inequalities. In [32,41,42], the authors used the Levin-Stečkin theorem [25] to study Hermite-Hadamard type inequalities. Many results on higher order generalizations of the Hermite-Hadamard type inequality one can found, among others, in [1,2,3,4,5,14,36,37].…”