Jensen-Mercer Inequality Research Articles

Inequalities of the Type Hermite–Hadamard–Jensen–Mercer for Strong Convexity

By using the Jensen–Mercer inequality for strongly convex functions, we present Hermite–Hadamard–Mercer inequality for strongly convex functions. Furthermore, we also present some new Hermite‐Hadamard‐Mercer-type inequalities for differentiable functions whose derivatives in absolute value are convex.

Generalized Fractal Jensen–Mercer and Hermite–Mercer type inequalities via h-convex functions involving Mittag–Leffler kernel

In this paper, we present generalized Jensen-Mercer inequality for a generalized h-convex function on fractal sets. We proved Hermite-Hadamard-Mercer local fractional integral inequalities via integral operators pertaining Mittag-Leffler kernel. Also, we drive two new local fractional integral identities for differentiable functions. By employing these integral identities, we derive some new Hermite-Mercer type inequalities for generalized h-convex function in local fractional calculus settings. Finally, we give some examples to emphasize the applications of derived results. These results will be a significant addition to Jensen-type inequalities in the literature.

New Fractional Hermite–Hadamard–Mercer Inequalities for Harmonically Convex Function

In 2003, Mercer presented an interesting variation of Jensen’s inequality called Jensen–Mercer inequality for convex function. In the present paper, by employing harmonically convex function, we introduce analogous versions of Hermite–Hadamard inequalities of the Jensen–Mercer type via fractional integrals. As a result, we introduce several related fractional inequalities connected with the right and left differences of obtained new inequalities for differentiable harmonically convex mappings. As an application viewpoint, new estimates regarding hypergeometric functions and special means of real numbers are exemplified to determine the pertinence and validity of the suggested scheme. Our results presented here provide extensions of others given in the literature. The results proved in this paper may stimulate further research in this fascinating area.

Generalizations of the Jensen–Mercer Inequality via Fink’s Identity

We generalize an integral Jensen–Mercer inequality to the class of n-convex functions using Fink’s identity and Green’s functions. We study the monotonicity of some linear functionals constructed from the obtained inequalities using the definition of n-convex functions at a point.

JENSEN–MERCER INEQUALITY AND RELATED RESULTS IN THE FRACTAL SENSE WITH APPLICATIONS

The most notable inequality pertaining convex functions is Jensen’s inequality which has tremendous applications in several fields. Mercer introduced an important variant of Jensen’s inequality called as Jensen–Mercer’s inequality. Fractal sets are useful tools for describing the accuracy of inequalities in convex functions. The purpose of this paper is to establish a generalized Jensen–Mercer inequality for a generalized convex function on a real linear fractal set [Formula: see text] ([Formula: see text]. Further, we also demonstrate some generalized Jensen–Mercer-type inequalities by employing local fractional calculus. Lastly, some applications related to Jensen–Mercer inequality and [Formula: see text]-type special means are given. The present approach is efficient, reliable, and may motivate further research in this area.

Generalization and refinements of the Jensen-Mercer inequality with applications

Generalization and refinements of the Jensen-Mercer inequality with applications

New Variant of Hermite–Jensen–Mercer Inequalities via Riemann–Liouville Fractional Integral Operators

In this paper, certain Hermite–Hadamard–Mercer-type inequalities are proved via Riemann–-Liouville fractional integral operators. We established several new variants of Hermite–Hadamard’s inequalities for Riemann–Liouville fractional integral operators by utilizing Jensen–Mercer inequality for differentiable mappingϒwhose derivatives in the absolute values are convex. Moreover, we construct new lemmas for differentiable functionsϒ′,ϒ″, andϒ‴and formulate related inequalities for these differentiable functions using variants of Hölder’s inequality.

A counterpart to Jensen-Mercer inequality

The main goal of this paper is to point out some refinements of the reverse of the Jensen-Mercer inequality.

The Jensen-Mercer Inequality with Infinite Convex Combinations

The paper deals with discrete forms of double inequalities related to convex functions of one variable. Infinite convex combinations and sequences of convex combinations are included. The double inequality form of the Jensen-Mercer inequality and its variants are especially studied.

Note on generalization of the Jensen-Mercer inequality by Taylor's polynomial

Note on generalization of the Jensen-Mercer inequality by Taylor's polynomial

Operator m-convex functions

Abstract The aim of this paper is to present a comprehensive study of operator m-convex functions. Let m ∈ [ 0 , 1 ] {m\in[0,1]} , and J = [ 0 , b ] {J=[0,b]} for some b ∈ ℝ {b\in\mathbb{R}} or J = [ 0 , ∞ ) {J=[0,\infty)} . A continuous function φ : J → ℝ {\varphi\colon J\to\mathbb{R}} is called operator m-convex if for any t ∈ [ 0 , 1 ] {t\in[0,1]} and any self-adjoint operators A , B ∈ 𝔹 ⁢ ( ℋ ) {A,B\in\mathbb{B}({\mathscr{H}})} , whose spectra are contained in J, we have φ ⁢ ( t ⁢ A + m ⁢ ( 1 - t ) ⁢ B ) ≤ t ⁢ φ ⁢ ( A ) + m ⁢ ( 1 - t ) ⁢ φ ⁢ ( B ) {\varphi(tA+m(1-t)B)\leq t\varphi(A)+m(1-t)\varphi(B)} . We first generalize the celebrated Jensen inequality for continuous m-convex functions and Hilbert space operators and then use suitable weight functions to give some weighted refinements. Introducing the notion of operator m-convexity, we extend the Choi–Davis–Jensen inequality for operator m-convex functions. We also present an operator version of the Jensen–Mercer inequality for m-convex functions and generalize this inequality for operator m-convex functions involving continuous fields of operators and unital fields of positive linear mappings. Employing the Jensen–Mercer operator inequality for operator m-convex functions, we construct the m-Jensen operator functional and obtain an upper bound for it.

Generalization of the Jensen-Mercer inequality by Taylor's polynomial

Generalization of the Jensen-Mercer inequality by Taylor's polynomial

Cauchy-type means and exponential and logarithmic convexity for superquadratic functions on time scales

n this paper, we define positive functionals by using the Jensen's inequality, converse of Jensen's inequality, and Jensen-Mercer's inequality on time scales for superquadratic functions. We give mean-value theorems and introduce related Cauchy-type means by using the functionals mentioned above and show the monotonicity of these means. We also show that these functionals are exponentially convex and give some applications of them by using the log-convexity and exponential convexity.

Majorization and refined Jensen–Mercer type inequalities for self-adjoint operators

In this paper, Jensen–Mercer's inequality is generalized by applying the method of pre-majorization used for comparing two tuples of self-adjoint operators. A general result in a matrix setting is established. Special cases of the main theorem are studied to recover other inequalities of Mercer type.

Inequalities of Convex Functions and Self-Adjoint Operators

The paper offers generalizations of the Jensen-Mercer inequality for self-adjoint operators and generally convex functions. The obtained results are applied to define the quasi-arithmetic operator means without using operator convexity. The version of the harmonic-geometric-arithmetic operator mean inequality is derived as an example.

On the refinements of Jensen Mercer's inequality

In this paper we give refinements of Jensen-Mercer's inequality and its generalizations and give applications for means. We prove \(n\)-exponential convexity of the functions constructed from these refinements. At the end we discuss some examples.

On a variant of the Jensen-Mercer inequality for operators

Some refinements of the Jensen-Mercer inequality for operators are presented. Ob- tained results are used to refine some comparision inequalities between power and quasi- arithmetic means for operators.

Refinements of Jensen-Mercer's inequality for index set functions with applications

Some refinements of Jensen-Mercer's inequality are presented. They are used to refine few inequalities among various means of Mercer's type, and they are further generalized for linear functionals.