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In this paper, several applications of the Hermite-Hadamard inequality for fractional integrals using harmonic convexity are discussed, including some new refinements and similar extensions, as well as several applications in the Beta function. [ABSTRACT FROM AUTHOR]

In this paper, we use the properties of Atangana–Baleanu (AB) fractional calculus and Prabhakar fractional calculus to construct some novel Hermite–Hadamard-type fractional integral inequalities for harmonically convex functions. And these inequalities are represented by the Mittag-Leffler functions. Finally, several special inequalities are established to illustrate the applications of our conclusions in special means. [ABSTRACT FROM AUTHOR]

In this paper, new Fejér-type inequalities for harmonically convex functions are obtained. Some mappings related to the Fejér-type inequalities for harmonically convex are defined. Properties of these mappings are discussed and, as a consequence, we obtain refinements of some known results. [ABSTRACT FROM AUTHOR]

In this paper, using positive symmetric functions, we offer two new important identities of fractional integral form for convex and harmonically convex functions. We then prove new variants of the Hermite-Hadamard-Fejér type inequalities for convex as well as harmonically convex functions via fractional integrals involving an exponential kernel. Moreover, we also present improved versions of midpoint type Hermite-Hadamard inequality. Graphical representations are given to validate the accuracy of the main results. Finally, applications associated with matrices, q-digamma functions and modifed Bessel functions are also discussed.

In this work, we prove Ostrowski–Mercer inequalities for differentiable harmonically convex functions. It is also shown that the newly proved inequalities can be converted into some existing inequalities. Furthermore, it is provided that how the newly discovered inequalities can be applied to special means of real numbers. [ABSTRACT FROM AUTHOR]

The pivotal proposal of this work is to present a new class of harmonically convex functions, namely, generalized harmonically ψ‐convex functions based on the fractal set technique for establishing inequalities of Hermite–Hadamard type, Pachpatte type, and certain related variants with respect to the Raina's function. With the aid of an auxiliary identity correlated with Raina's function, by generalized Hölder inequality, two Hermite–Hadamard‐type local fractional integral inequalities for generalized harmonically ψ‐convex functions are apprehended. The proposed technique provides the results by giving some special values to the parameters or imposing restrictive assumptions and is completely feasible for recapturing the existing results in the relative literature. To determine the computational efficiency of offered scheme, some numerical applications are discussed. The results of the scheme show that the approach is straightforward to apply and computationally very user‐friendly and accurate. [ABSTRACT FROM AUTHOR]

Integral inequalities have accumulated a comprehensive and prolific field of research within mathematical interpretations. In recent times, strategies of fractional calculus have become the subject of intensive research in historical and contemporary generations because of their applications in various branches of science. In this paper, we concentrate on establishing Hermite–Hadamard and Pachpatte-type integral inequalities with the aid of two different fractional operators. In particular, we acknowledge the critical Hermite–Hadamard and related inequalities for n-polynomial s-type convex functions and n-polynomial s-type harmonically convex functions. We practice these inequalities to consider the Caputo–Fabrizio and the k-Riemann–Liouville fractional integrals. Several special cases of our main results are also presented in the form of corollaries and remarks. Our study offers a better perception of integral inequalities involving fractional operators. [ABSTRACT FROM AUTHOR]

In this paper, new Fejér-type inequalities for harmonically convex functions are obtained. Some mappings related to the Fejér-type inequalities for harmonically convex are defined. Properties of these mappings are discussed and, as a consequence, we obtain refinements of some known results.

Abstract The visual beauty reflects the practicability and superiority of design dependent on the fractal theory. Based on the applicability in practice, it shows that it is the completely feasible, self-comparability and multifaceted nature of fractal sets that made it an appealing field of research. There is a strong correlation between fractal sets and convexity due to its intriguing nature in the mathematical sciences. This paper investigates the notions of generalized exponentially harmonically ( G E H $GEH$ ) convex and G E H $GEH$ s-convex functions on a real linear fractal sets R α $\mathbb{R}^{\alpha }$ ( 0 < α ≤ 1 $0

In this paper, we introduce the concept of right semi-harmonically (γ , η) -convex, (RHS (γ , η) -convex) function by the inequality f (γ x , y (t)) ≤ 1 η 1 f (x) , 1 f (y) (t) and also left semi-harmonically (γ , η) -convex, (LHS (γ , η) -convex) f 1 γ 1 y , 1 x (t) ≤ η f (y) , f (x) (t) where γ and η are two geodesic arcs. Then, we will find some refinements of Hadamard integral inequality for semi-harmonically (γ , η) -convex functions in the case of Sugeno and Lebesque integral. [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is to establish new generalized fractional versions of the Hadamard and the Fejér–Hadamard integral inequalities for harmonically convex functions. Fractional integral operators involving an extended generalized Mittag-Leffler function which are further generalized via a monotone increasing function are utilized to get these generalized fractional versions. The results of this paper give several consequent fractional inequalities for harmonically convex functions for known fractional integral operators deducible from utilized generalized fractional integral operators.

In this paper, a new identity is established for differentiable mappings. By using the mathematical analysis techniques, some new integral inequalities of Hermite-Hadamard type for differentiable p-convex functions are proved. A comparison of the established results with previously obtained results is demonstrated to show that the results presented in this paper are better than those already exist in literature. [ABSTRACT FROM AUTHOR]

Integral inequalities have accumulated a comprehensive and prolific field of research within mathematical interpretations. In recent times, strategies of fractional calculus have become the subject of intensive research in historical and contemporary generations because of their applications in various branches of science. In this paper, we concentrate on establishing Hermite–Hadamard and Pachpatte-type integral inequalities with the aid of two different fractional operators. In particular, we acknowledge the critical Hermite–Hadamard and related inequalities for n-polynomial s-type convex functions and n-polynomial s-type harmonically convex functions. We practice these inequalities to consider the Caputo–Fabrizio and the k-Riemann–Liouville fractional integrals. Several special cases of our main results are also presented in the form of corollaries and remarks. Our study offers a better perception of integral inequalities involving fractional operators.

In this article, we established a new version of generalized fractional Hadamard and Fejér--Hadamard type integral inequalities. A fractional integral operator (FIO) with a non-singular function (multi-index Bessel function) as its kernel and monotone increasing functions is utilized to obtain the new version of such fractional inequalities. Our derived results are a generalized form of several proven inequalities already existing in the literature. The proven inequalities are useful for studying the stability and control of corresponding fractional dynamic equations. [ABSTRACT FROM AUTHOR]

For the classical Hermite-Hadamard inequality of harmonically convex functions, i.e., ... an upper bound is proved in the framework of the Sugeno integral. [ABSTRACT FROM AUTHOR]

The visual beauty reflects the practicability and superiority of design dependent on the fractal theory. Based on the applicability in practice, it shows that it is the completely feasible, self-comparability and multifaceted nature of fractal sets that made it an appealing field of research. There is a strong correlation between fractal sets and convexity due to its intriguing nature in the mathematical sciences. This paper investigates the notions of generalized exponentially harmonically (G E H ) convex and G E H s-convex functions on a real linear fractal sets R α (0 < α ≤ 1 ). Based on these novel ideas, we derive the generalized Hermite–Hadamard inequality, generalized Fejér–Hermite–Hadamard type inequality and Pachpatte type inequalities for G E H s-convex functions. Taking into account the local fractal identity; we establish a certain generalized Hermite–Hadamard type inequalities for local differentiable G E H s-convex functions. Meanwhile, another auxiliary result is employed to obtain the generalized Ostrowski type inequalities for the proposed techniques. Several special cases of the proposed concept are presented in the light of generalized exponentially harmonically convex, generalized harmonically convex and generalized harmonically s-convex. Meanwhile, an illustrative example and some novel applications in generalized special means are obtained to ensure the correctness of the present results. This novel strategy captures several existing results in the corresponding literature. Finally, we suppose that the consequences of this paper can stimulate those who are interested in fractal analysis. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish new generalized fractional versions of the Hadamard and the Fejér–Hadamard integral inequalities for harmonically convex functions. Fractional integral operators involving an extended generalized Mittag-Leffler function which are further generalized via a monotone increasing function are utilized to get these generalized fractional versions. The results of this paper give several consequent fractional inequalities for harmonically convex functions for known fractional integral operators deducible from utilized generalized fractional integral operators. [ABSTRACT FROM AUTHOR]

Present article takes into account results on Hadamard type inequality. It intents to generalize some of the existing fractional integral inequalities of Hadamard type inequality for harmonically convex functions to corresponding k-fractional inequalities, by using the classic k-fractional integrals. For this purpose first a significant Hadamard type inequality for harmonically convex functions via k-fractional integrals is procured. Then the differences of middle term with right and left terms of the aforesaid inequalities are estimated. [ABSTRACT FROM AUTHOR]

In this paper, we prove Hermite–Hadamard–Mercer inequalities, which is a new version of the Hermite–Hadamard inequalities for harmonically convex functions. We also prove Hermite–Hadamard–Mercer-type inequalities for functions whose first derivatives in absolute value are harmonically convex. Finally, we discuss how special means can be used to address newly discovered inequalities.

In this article, we established a new version of generalized fractional Hadamard and Fejér–Hadamard type integral inequalities. A fractional integral operator (FIO) with a non-singular function (multi-index Bessel function) as its kernel and monotone increasing functions is utilized to obtain the new version of such fractional inequalities. Our derived results are a generalized form of several proven inequalities already existing in the literature. The proven inequalities are useful for studying the stability and control of corresponding fractional dynamic equations.

In this paper, a new weighted identity involving a differentiable mapping and a non-negative p-symmetric mapping is established. By using the mathematical analysis techniques, some new integral inequalities of Hermite-Hadamard and Fejer type for differentiable p-convex functions are proved. A comparison of the established results is presented with the help of suitable graphs by using the software Mathematica. [ABSTRACT FROM AUTHOR]

The object of this paper is to obtain certain Hermite-Hadamard type integral inequalities involving general class of fractional integral operators and the fractional integral operators with exponential kernel by using harmonically convex functions. [ABSTRACT FROM AUTHOR]

In this work, we pose a new equality for differentiable functions. By using this equality, we have some new Ostrowki type inequalities and some error estimates for the midpoint formula for functions whose derivatives in absolute values at certain powers are harmonically convex. [ABSTRACT FROM AUTHOR]

By using an identity involving a twice differentiable mapping and mathematical analysis techniques, some new estimates are presented for the error bounds of... We have derived some inequalities of special menas of positive real numbers as applications of the proven results. [ABSTRACT FROM AUTHOR]

The decision theorem of the Schur harmonic convexity for the compositions involving Schur harmonically convex functions is established and used to determine the Schur harmonic convexity of some symmetric functions. [ABSTRACT FROM AUTHOR]

In this paper, firstly, Hermite-Hadamard-Fejér type inequality for harmonically convex functions in fractional integral forms have been established. Secondly, an integral identity and some Hermite-Hadamard-Fejér type integral inequalities for harmonically convex functions in fractional integral forms have been obtained. The some results presented here would provide extensions of those given in earlier works. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the authors established Hermite–Hadamard’s inequalities for harmonically convex functions via fractional integrals and obtained some Hermite–Hadamard type inequalities of these classes of functions. [Copyright &y& Elsevier]

Fifteen years ago, J. Borwein, I. Affleck, and R. Girgensohn posed a problem concerning the shape (convexity, log-convexity, reciprocal concavity) of a certain function of several arguments that had manifested in a number of contexts concerned with optimization problems. In this paper we further explore the shape of the Borwein-Affleck-Girgensohn function as well as of its extensions generated by completely monotone and Bernstein functions. [ABSTRACT FROM AUTHOR]

In this paper, we prove Hermite–Hadamard–Mercer inequalities, which is a new version of the Hermite–Hadamard inequalities for harmonically convex functions. We also prove Hermite–Hadamard–Mercer-type inequalities for functions whose first derivatives in absolute value are harmonically convex. Finally, we discuss how special means can be used to address newly discovered inequalities. [ABSTRACT FROM AUTHOR]