Hermite-Hadamard Inequalities Research Articles

Generalization of the Fuzzy Fejér–Hadamard Inequalities for Non-Convex Functions over a Rectangle Plane

Integral inequalities with generalized convexity play a vital role in both theoretical and applied mathematics. The theory of integral inequalities is one of the branches of mathematics that is now developing at the quickest rate due to its wide range of applications. We define a new Hermite–Hadamard inequality for the novel class of coordinated ƛ-pre-invex fuzzy number-valued mappings (C-ƛ-pre-invex 𝘍 𝘕 𝘝 𝘔s) and examine the idea of C-ƛ-pre-invex 𝘍 𝘕 𝘝 𝘔s in this paper. Furthermore, using C-ƛ-pre-invex 𝘍 𝘕 𝘝 𝘔s, we construct several new integral inequalities for fuzzy double Riemann integrals. Several well-known results, as well as recently discovered results, are included in these findings as special circumstances. We think that the findings in this work are new and will help to stimulate more research in this area in the future. Additionally, unique choices lead to new outcomes.

Improved Hermite–Hadamard Inequality Bounds for Riemann–Liouville Fractional Integrals via Jensen’s Inequality

Improved Hermite–Hadamard Inequality Bounds for Riemann–Liouville Fractional Integrals via Jensen’s Inequality

Fractional Hermite–Hadamard, Newton–Milne, and Convexity Involving Arithmetic–Geometric Mean-Type Inequalities in Hilbert and Mixed-Norm Morrey Spaces ℓq(·)(Mp(·),v(·)) with Variable Exponents

Function spaces play a crucial role in the study and application of mathematical inequalities. They provide a structured framework within which inequalities can be formulated, analyzed, and applied. They allow for the extension of inequalities from finite-dimensional spaces to infinite-dimensional contexts, which is crucial in mathematical analysis. In this note, we develop various new bounds and refinements of different well-known inequalities involving Hilbert spaces in a tensor framework as well as mixed Moore norm spaces with variable exponents. The article begins with Newton–Milne-type inequalities for differentiable convex mappings. Our next step is to take advantage of convexity involving arithmetic–geometric means and build various new bounds by utilizing self-adjoint operators of Hilbert spaces in tensorial frameworks for different types of generalized convex mappings. To obtain all these results, we use Riemann–Liouville fractional integrals and develop several new identities for these operator inequalities. Furthermore, we present some examples and consequences for transcendental functions. Moreover, we developed the Hermite–Hadamard inequality in a new and significant way by using mixed-norm Moore spaces with variable exponent functions that have not been developed previously with any other type of function space apart from classical Lebesgue space. Mathematical inequalities supporting tensor Hilbert spaces are rarely examined in the literature, so we believe that this work opens up a whole new avenue in mathematical inequality theory.

Bounds of Different Integral Operators in Tensorial Hilbert and Variable Exponent Function Spaces

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of these operators is crucial. Functional analysis typically involves the use of tensors to represent multilinear mappings between Hilbert spaces, which can result in inequality in tensor Hilbert spaces. In this paper, we study two types of function spaces and use convex and harmonic convex mappings to establish various operator inequalities and their bounds. In the first part of the article, we develop the operator Hermite–Hadamard and upper and lower bounds for weighted discrete Jensen-type inequalities in Hilbert spaces using some relational properties and arithmetic operations from the tensor analysis. Furthermore, we use the Riemann–Liouville fractional integral and develop several new identities which are used in operator Milne-type inequalities to develop several new bounds using different types of generalized mappings, including differentiable, quasi-convex, and convex mappings. Furthermore, some examples and consequences for logarithm and exponential functions are also provided. Furthermore, we provide an interesting example of a physics dynamical model for harmonic mean. Lastly, we develop Hermite–Hadamard inequality in variable exponent function spaces, specifically in mixed norm function space (lq(·)(Lp(·))). Moreover, it was developed using classical Lebesgue space (Lp) space, in which the exponent is constant. This inequality not only refines Jensen and triangular inequality in the norm sense, but we also impose specific conditions on exponent functions to show whether this inequality holds true or not.

Generalization of quantum calculus and corresponding Hermite–Hadamard inequalities

The aim of this paper is first to introduce generalizations of quantum integrals and derivatives which are called (ϕ-h)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\phi \\,-\\,h)$$\\end{document} integrals and (ϕ-h)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\phi \\,-\\,h)$$\\end{document} derivatives, respectively. Then we investigate some implicit integral inequalities for (ϕ-h)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\phi \\,-\\,h)$$\\end{document} integrals. Different classes of convex functions are used to prove these inequalities for symmetric functions. Under certain assumptions, Hermite–Hadamard-type inequalities for q-integrals are deduced. The results presented herein are applicable to convex, m-convex, and ħ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar $$\\end{document}-convex functions defined on the non-negative part of the real line.

On a weighted Hermite–Hadamard inequality involving convex functional arguments

On a weighted Hermite–Hadamard inequality involving convex functional arguments

Right Quantum Calculus on Finite Intervals with Respect to Another Function and Quantum Hermite–Hadamard Inequalities

Right Quantum Calculus on Finite Intervals with Respect to Another Function and Quantum Hermite–Hadamard Inequalities

Hermite-Hadamard Inequalities for a New Class of Generalized-(s,m) via Fractional Integral

This paper defines a new generalized (s,m)-σ convex function using the σ convex functions and provides some applications and exact results for this kind of functions. The new definition of the (s,m)-σ convex function class is used to obtain the Hermite Hadamard type integral inequalities existing in the literature, and new integral inequalities are obtained with the help of the σ-Riemann-Liouville fractional integral. Additionally, a new Hermite-Hadamard type fractional integral inequality is constructed using the σ-Riemann-Liouville fractional integral.

Hermite-Hadamard inequalities for generalized (m-F)-convex function in the framework of local fractional integrals

This work presents new versions of the Hermite-Hadamard Inequality, for (m−F)-convex functions, defined on fractal sets Rς ( 0 ς ≤ 1). So, we show some new results for twice differentiable functions using local fractional calculus, as well as some new definitions. We will construct these new integral inequality using the generalized H¨older-integral inequality and the power mean integral inequality. Furthermore, we present some new inequalities for the midpoint and trapezoid formulas in a novel type of fractal calculus. The conclusions in this paper are substantial advancements and generalizations of prior research reported in the literature.

Weighted fractional inequalities for new conditions on h-convex functions

We use a new function class called B-function to establish a novel version of Hermite–Hadamard inequality for weighted ψ-Hilfer operators. Additionally, we prove two new identities involving weighted ψ-Hilfer operators for differentiable functions. Moreover, by employing these equalities and the properties of the B-function, we derive several trapezoid- and midpoint-type inequalities for h-convex functions. Furthermore, the obtained results are reduced to several well-known and some new inequalities by making specific choices of the function h.

Some Inequalities for Differentiable Functions on a Generalization of Hermite-Hadamard Inequality with Applications

The main identity of midpoint type inequalities is generalized and using this identity some composite midpoint type inequalities are estimated. Moreover, applications to some special means and some error estimates for the composite midpoint formula are discussed.

Better Approximations for Quasi-Convex Functions

In this paper, by using Hölder-İşcan, Hölder integral inequality and a general identity for differentiable functions we can get new estimates on generalization of Hadamard, Ostrowski and Simpson type integral inequalities for functions whose derivatives in absolute value at certain power are quasi-convex functions. It is proved that the result obtained Hölder-İşcan integral inequality is better than the result obtained Hölder inequality. Keywords: Hölder-İşcan scan inequality, Hermite-Hadamard inequality, Simpson and Ostrowski type inequality, midpoint and trapezoid type inequality, quasi-convex functions.

Symmetric Quantum Inequalities on Finite Rectangular Plane

Finding the range of coordinated convex functions is yet another application for the symmetric Hermite–Hadamard inequality. For any two-dimensional interval [a0,a1]×[c0,c1]⊆ℜ2, we introduce the notion of partial qθ-, qϕ-, and qθqϕ-symmetric derivatives and a qθqϕ-symmetric integral. Moreover, we will construct the qθqϕ-symmetric Hölder’s inequality, the symmetric quantum Hermite–Hadamard inequality for the function of two variables in a rectangular plane, and address some of its related applications.

Exploring nine different proofs of a famous logarithmic inequality

This article explores and analyzes nine different proofs of a fundamental logarithmic inequality, say "log(1+x)/x ≥ 1/ (1+x/2) for x>-1". Some proofs are already published; others are new or consider new approaches or new angles of research. They are based on various techniques, such as differentiation, series expansion, and the application of well-known inequalities such as the primitive, Cauchy-Schwarz, logarithmic mean, hyperbolic tangent function, Jensen, and Hermite-Hadamard inequalities. A graphical work illustrates some results. Therefore, our study clarifies the different mathematical foundations of this seemingly straightforward but important inequality and goes beyond it with some new material.

Hyers–Ulam Stability of 2D-Convex Mappings and Some Related New Hermite–Hadamard, Pachpatte, and Fejér Type Integral Inequalities Using Novel Fractional Integral Operators via Totally Interval-Order Relations with Open Problem

The aim of this paper is to introduce a new type of two-dimensional convexity by using total-order relations. In the first part of this paper, we examine the Hyers–Ulam stability of two-dimensional convex mappings by using the sandwich theorem. Our next step involves the development of Hermite–Hadamard inequality, including its weighted and product forms, by using a novel type of fractional operator having non-singular kernels. Moreover, we develop several nontrivial examples and remarks to demonstrate the validity of our main results. Finally, we examine approximate convex mappings and have left an open problem regarding the best optimal constants for two-dimensional approximate convexity.

Bullen-Mercer type inequalities with applications in numerical analysis

In mathematical analysis theory of inequalities has considerable influence due to its massive utility in various fields of physical sciences. These are investigated via multiple approaches to acquire more precise and rectified forms of already celebrated consequences. Integral inequalities are investigated to compute the error bounds for quadrature schemes. Among all of them, one is Hermite-Hadamard inequality, which has mighty efficacy. Numerous generalizations have been proposed in the literature based on different novel and innovative procedures. In recent years, Bullen inequality has been very commonly studied inequality. The main objective of our progressive study is to establish a new set of Bullen-type inequalities concerning the Jensen-Mecer inequality. For the completion of the current investigation, we derive a new general Bullen-Mecer equality, which is beneficial to achieve our primary consequences. Furthermore, Considering the Bullen-Mecer equation, we employ the convexity property together with famous Hölder's type and Young's inequalities, bounding, and Lipschitz characteristics of functions to conclude new variants of generalized upper bounds of Bullen inequality. Also, we deliver some applications of outcomes to means, special functions, error bounds, and iterative methods to solve non-linear problems. Lastly, we verify our findings through various simulations. The advantage of the current study is that several results concerning Bullen's inequality can be retrieved from our proposed results and various new results can be achieved by choosing the values for γ and δ. By utilizing the similar technique that we have adopted new iterative schemes can be established from integral inequalities.

A novel class of integral inequalities with graphical approach and diverse applications

ABSTRACT In this paper, we investigate a novel class of Hermite-Hadamard inequalities applicable to functions with h-convex absolute derivatives. Graphical representations are provided to bolster the validity of our key findings. Some limiting results of our main findings are discussed as corollaries. Furthermore, we establish error estimates in terms of trapezoid formulae for differences between generalized means.

Generalized strongly n-polynomial convex functions and related inequalities

This paper focuses on introducing and examining the class of generalized strongly n-polynomial convex functions. Relationships between these functions and other types of convex functions are explored. The Hermite–Hadamard inequality is established for generalized strongly n-polynomial convex functions. Additionally, new integral inequalities of Hermite–Hadamard type are derived for this class of functions using the Hölder–İşcan integral inequality. The results obtained in this paper are compared with those known in the literature, demonstrating the superiority of the new results. Finally, some applications for special means are provided.

New Version of Fractional Pachpatte-Type Integral Inequalities via Coordinated ℏ-Convexity via Left and Right Order Relation

In particular, the fractional forms of Hermite–Hadamard inequalities for the newly defined class of convex mappings proposed that are known as coordinated left and right ℏ-convexity (LR-ℏ-convexity) over interval-valued codomain. We exploit the use of double Riemann–Liouville fractional integral to derive the major results of the research. We also examine the key results’ numerical validations that examples are nontrivial. By taking the product of two left and right coordinated ℏ-convexity, some new versions of fractional integral inequalities are also obtained. Moreover, some new and classical exceptional cases are also discussed by taking some restrictions on endpoint functions of interval-valued functions that can be seen as applications of these new outcomes.

New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions

Abstract In this study, based on two new local fractional integral operators involving generalized Mittag-Leffler kernel, Hermite-Hadamard inequality about these two integral operators for generalized h h -preinvex functions is obtained. Subsequently, an integral identity related to these two local fractional integral operators is constructed to obtain some new Ostrowski-type local fractional integral inequalities for generalized h h -preinvex functions. Finally, we propose three examples to illustrate the partial results and applications. Meanwhile, we also propose two midpoint-type inequalities involving generalized moments of continuous random variables to show the application of the results.