Some Hermite–Hadamard type inequalities for functions of generalized convex derivative

We obtain some new inequalities of Hermite–Hadamard type. We consider functions that have convex or generalized convex derivative. Additional inequalities are proven for functions whose second derivative in absolute values are convex. Applications of the main results are presented.

     

Hermite–Hadamard inequality for convex stochastic processes

In this note we extend the classical Hermite–Hadamard inequality to convex stochastic processes.     

Hermite–Hadamard’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces

Some Hermite–Hadamard’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications in relation with the celebrated Hölder–McCarthy’s inequality for positive operators and Ky Fan’s inequality for real numbers are given as well.     

Hermite–Hadamard type inequalities for the m- and (α, m)-geometrically convex functions

In the paper the authors introduce concepts of the m- and (α, m)-geometrically convex functions and establish some inequalities of Hermite–Hadamard type for these classes of functions.     

Some Hermite–Hadamard type inequalities for geometrically quasi-convex functions

In the paper, we introduce a new concept ‘geometrically quasi-convex function’ and establish some Hermite–Hadamard type inequalities for functions whose derivatives are of geometric quasi-convexity.     

New Hermite–Hadamard-type inequalities for convex functions (I)

In this work we establish some new Hermite–Hadamard-type inequalities for convex functions and give several applications for special means.     

A functional inequality characterizing convex functions,conjugacy and a generalization of Hölder's and Minkowski's inequalities

Summary The main result says that, iff: _{+ +} satisfies the functional inequalityaf(s) + bf(t) f (as + bt) (s,t 0) for somea, b such that 0 <a < 1 <a + b, thenf(t) = f(1)t, (t 0). A relevant result for the reverse inequality is also discussed. Applying these results we determine the form of all functionsf: _k ⁺ ₊ satisying the above inequalities. This leads to a characterization of both concave and convex functions defined on ₊ ^k–1 , to a notion of conjugate functions and to a general inequality which contains Hölder's and Minkowski's inequalities as very special cases.     

On fractional stochastic inequalities related to Hermite–Hadamard and Jensen types for convex stochastic processes

Stochastic convexity and its applications are very important in mathematics and probability (Aequationes Mathematicae 20:184–197, 1980). There are two well-known inequalities for convex stochastic processes: Jensen’s inequality and Hermite–Hadamard’s inequality. Recently, Hafiz (Stoch Anal Appl 22:507–523, 2004) has provided fractional calculus for some stochastic processes. The problem is how to formulate these inequalities for stochastic processes in the class of fractional calculus and that is what is done in this paper. Our results generalize the corresponding ones in the literature.     

A refinement of the right-hand side of the Hermite–Hadamard inequality for simplices

Aequationes mathematicae - We establish a new refinement of the right-hand side of the Hermite–Hadamard inequality for simplices, based on the average values of a convex function over the...     

Fejér and Hermite–Hadamard type results for H-invex functions with applications

Positivity - We investigate the class of H-invex functions including, e.g., the subclasses of convex, c-strongly convex, $$ \varphi $$ -uniformly convex and superquadratic functions. For H-invex...     

Korovkin type theorems and approximate Hermite–Hadamard inequalities

The main results of this paper offer sufficient conditions in order that an approximate lower Hermite–Hadamard type inequality implies an approximate convexity property. The failure of such an implication with constant error term shows that functional error terms should be considered for the inequalities and convexity properties in question. The key for the proof of the main result is a Korovkin type theorem which enables us to deduce the approximate convexity property from the approximate lower Hermite–Hadamard type inequality via an iteration process.     

On symmetrized stochastic convexity and the inequalities of Hermite–Hadamard type

Aequationes mathematicae - In this paper the concept of symmetrized convex stochastic processes is introduced. Some characterizations involving Hermite–Hadamard type inequalities and a...     

Another refinement of the right-hand side of the Hermite–Hadamard inequality for simplices

We establish a new refinement of the right-hand side of the Hermite–Hadamard inequality for convex functions of several variables defined on simplices.     

The converse of the Hermite–Hadamard inequality on simplices

We prove that the Hermite–Hadamard inequality on simplices characterizes convex functions under some assumptions on the measure.     

A refined Brunn–Minkowski inequality for convex sets

Starting from a mass transportation proof of the Brunn–Minkowski inequality on convex sets, we improve the inequality showing a sharp estimate about the stability property of optimal sets. This is based on a Poincaré-type trace inequality on convex sets that is also proved in sharp form.     

Refinements of Fejér’s inequality for convex functions

In this paper, we establish some new refinements for the celebrated Fejér??s and Hermite-Hadamard??s integral inequalities for convex functions.