Higher Order Convex Functions Research Articles

Hermite–Hadamard type inequalities for higher-order convex functions and applications

Hermite–Hadamard type inequalities for higher-order convex functions and applications

Some Interesting Inequalities for the Class of Generalized Convex Functions of Higher Order

Some Interesting Inequalities for the Class of Generalized Convex Functions of Higher Order

Hermite–Hadamard and Fejér-type inequalities for strongly reciprocally (p, h)-convex functions of higher order

In this paper, we investigate the properties of a newly introduced class of functions, strongly reciprocally (p, h)-convex functions of higher order. We establish Hermite–Hadamard-type and Fejér-type inequalities for this class of functions. Additionally, we present fractional integral inequalities applicable to strongly reciprocally (p, h)-convex functions of higher order.

Some new integral inequalities for higher-order strongly exponentially convex functions

Integral inequalities with generalized convexity play an important role in both applied and theoretical mathematics. The theory of integral inequalities is currently one of the most rapidly developing areas of mathematics due to its wide range of applications. In this paper, we study the concept of higher-order strongly exponentially convex functions and establish a new Hermite–Hadamard inequality for the class of strongly exponentially convex functions of higher order. Further, we derive some new integral inequalities for Riemann–Liouville fractional integrals via higher-order strongly exponentially convex functions. These findings include several well-known results and newly obtained results as special cases. We believe that the results presented in this paper are novel and will be beneficial in encouraging future research in this field.

Hermite–Hadamard-Type Inequalities and Two-Point Quadrature Formula

As convexity plays an important role in many aspects of mathematical programming, e.g., for obtaining sufficient optimality conditions and in duality theorems, and one of the most important inequalities for convex functions is the Hermite–Hadamard inequality, the importance of this paper lies in providing some new improvements for convex functions and new directions in studying new variants of the Hermite–Hadamard inequality. The first part of the article includes some known concepts regarding convex functions and related inequalities. In the second part of the study, a derivation of the Hermite–Hadamard inequality for convex functions of higher order is given, emphasizing the purpose and importance of some quadrature formulas. In the third section, the applications of the main results are presented by obtaining Hermite–Hadamard-type estimates for various classical quadrature formulas such as the Gauss–Legendre two-point quadrature formula and the Gauss–Chebyshev two-point quadrature formulas of the first and second kind.

A Note on Generalized Strongly p-Convex Functions of Higher Order

Generalized strongly -convex functions of higher order is a new concept of convex functions which introduced by Saleem et al. in 2020. The Schur type inequality for generalized strongly -convex functions of higher order also studied by them. This paper aims to revise Schur type inequality for generalized strongly -convex functions of higher order in their paper. In order to revise it, we show that the contradiction was true. This paper showed that Schur type inequality for generalized strongly -convex functions of higher order previously is not valid and we give the correct Schur type inequality for generalized strongly -convex functions of higher order

Positivity of sums and integrals for n-convex functions via the Fink identity and new Green functions

We consider positivity of sum $\sum_{i=1}^np_if(x_i)$ involving convex functions of higher order. Analogous for integral $\int_a^bp(x)f(g(x))dx$ is also given. Representation of a function $f$ via the Fink identity and the Green function leads us to identities for which we obtain conditions for positivity of the mentioned sum and integral. We obtain bounds for integral remainders which occur in those identities as well as corresponding mean value theorems.

New entropic bounds on time scales via Hermite interpolating polynomial

Hermite’s interpolation is utilized to establish a new generalization of an inequality for higher order convex functions containing Csiszár divergence on time scales. New entropic bounds in q-calculus and h-discrete calculus are also deduced. Some estimates for Zipf–Mandelbrot entropy are also given.

Estimation of divergence measures on time scales via Taylor\u2019s polynomial and Green\u2019s function with applications in q-calculus

Taylor’s polynomial and Green’s function are used to obtain new generalizations of an inequality for higher order convex functions containing Csiszár divergence on time scales. Various new inequalities for some divergence measures in quantum calculus and h-discrete calculus are also established.

Hermite–Hadamard–Fejér-Type Inequalities and Weighted Three-Point Quadrature Formulae

The goal of this paper is to derive Hermite–Hadamard–Fejér-type inequalities for higher-order convex functions and a general three-point integral formula involving harmonic sequences of polynomials and w-harmonic sequences of functions. In special cases, Hermite–Hadamard–Fejér-type estimates are derived for various classical quadrature formulae such as the Gauss–Legendre three-point quadrature formula and the Gauss–Chebyshev three-point quadrature formula of the first and of the second kind.

Fejér type inequalities for higher order convex functions and quadrature formulae

Fejér type inequalities for higher order convex functions and quadrature formulae

On the Ra\u015fa Inequality for Higher Order Convex Functions

We study the following (q-1)th convex ordering relation for qth convolution power of the difference of probability distributions mu and nu (ν-μ)∗q≥(q-1)cx0,q≥2,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} (\\nu -\\mu )^{*q}\\ge _{(q-1)cx} 0 , \\quad q\\ge 2, \\end{aligned}$$\\end{document}and we obtain the theorem providing a useful sufficient condition for its verification. We apply this theorem for various families of probability distributions and we obtain several inequalities related to the classical interpolation operators. In particular, taking binomial distributions, we obtain a new, very short proof of the inequality given recently by Abel and Leviatan (2020).

Simpson type inequalities and applications

A new generalized integral identity involving first order differentiable functions is obtained. Using this identity as an auxiliary result, we then obtain some new refinements of Simpson type inequalities using a new class called as strongly (s, m)-convex functions of higher order of sigma >0. We also discuss some interesting applications of the obtained results in the theory of means. In last we present applications of the obtained results in obtaining Simpson-like quadrature formula.

Bounds for the Remainder in Simpson’s Inequality vian-Polynomial Convex Functions of Higher Order Using Katugampola Fractional Integrals

The goal of this paper is to derive some new variants of Simpson’s inequality using the class ofn-polynomial convex functions of higher order. To obtain the main results of the paper, we first derive a new generalized fractional integral identity utilizing the concepts of Katugampola fractional integrals. This new fractional integral identity will serve as an auxiliary result in the development of the main results of this paper.

Strongly Convex Functions of Higher Order Involving Bifunction

Some new concepts of the higher order strongly convex functions involving an arbitrary bifuction are considered in this paper. Some properties of the higher order strongly convex functions are investigated under suitable conditions. Some important special cases are discussed. The parallelogram laws for Banach spaces are obtained as applications of higher order strongly affine convex functions as novel applications. Results obtained in this paper can be viewed as refinement and improvement of previously known results.

Levinson type inequalities for higher order convex functions via Abel\u2013Gontscharoff interpolation

In this paper, Levinson type inequalities are studied for the class of higher order convex functions by using Abel–Gontscharoff interpolation. Cebyšev, Grüss, and Ostrowski-type new bounds are also found for the functionals involving data points of two types.

Several new cyclic Jensen type inequalities and their applications

We present some fundamental results and definitions regarding Jensen’s inequality with the aim of obtaining new generalizations of cyclic refinements of Jensen’s inequality from convex to higher order convex functions using Taylor’s formula. We discuss the monotonicity of functionals for n-convex functions at a point. Applications of our work include new bounds for some important inequalities used in information theory.

On strongly generalized convex functions of higher order

On strongly generalized convex functions of higher order

Generalization of cyclic refinements of Jensen inequality by montgomery identity and Green’s function

We consider discrete and continuous cyclic refinements of Jensen’s inequality and generalize them from convex function to higher order convex function by means of Lagrange Green’s function and Montgomery identity. We give application of our results by formulating the monotonicity of the linear functionals obtained from generalized identities utilizing the theory of inequalities for [Formula: see text]-convex functions at a point. We compute Grüss and Ostrowski type bounds for generalized identities associated with the obtained inequalities. Finally, we investigate the properties of linear functionals regarding exponential convexity log convexity and mean value theorems.

Sherman\u2019s and related inequalities with applications in information theory

In this paper we give extensions of Sherman’s inequality considering the class of convex functions of higher order. As particular cases, we get an extended weighted majorization inequality as well as Jensen’s inequality which have direct connection to information theory. We use the obtained results to derive new estimates for Shannon’s and Rényi’s entropy, information energy, and some well-known measures between probability distributions. Using the Zipf–Mandelbrot law, we introduce new functionals to derive some related results.