Inequalities of the Hermite–Hadamard Type Involving Numerical Differentiation Formulas

Abstract

Abstract. We observe that the Hermite-Hadamard inequality written in

Functional Inequalities Involving Numerical Differentiation Formulas of Order Two

“…Observe that from the equality, we know that functions F 9 , F 6 must have a crossing point in the interval (0, 1 − α), further these functions cross also at the point 1 − α. Thus there are two crossing points of F 9 , F 6 in view of Lemma 2 from [9] this means that expressions are incomparable in the class of convex functions (as claimed). The reasoning in the case α > 2 3 is similar.…”

“…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.…”

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

“…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…”

“…is satisfied for every continuous and convex f and its antiderivative F. Example 6 ( [32]). Using (ii), we can see that the inequality…”

“…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].…”