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By virtue of the left-sided fractional integral operators having exponential kernels, proposed by Ahmad et al. in [J. Comput. Appl. Math. 353:120-129, 2019], we create the left-sided fractional Hermite–Hadamard type inequalities for convex mappings. Moreover, to study certain fractional trapezoid and midpoint type inequalities via the differentiable convex mappings, two fractional integral identities are proven. Also, we show the important connections of the derived outcomes with those classical integrals clearly. Finally, we provide three numerical examples to verify the correctness of the presented inequalities that occur with the variation of the parameter μ.

In this paper, we introduce a class of the multiplicative tempered fractional integral operators. Then, we investigate two Hermite–Hadamard type inequalities for this class. By using the established identity and the multiplicative convexity, we establish some integral inequalities for the multiplicative tempered fractional integrals involving the λ-incomplete gamma functions. And our results obtained in the present paper generalize some results given by Budak and Tunç (2020) and Ali et al. (2019). Also, we provide three examples to demonstrate the simplicities of the calculations.

We use the definition of a fractional integral operators, recently introduced by Katugampola, to establish a parameterized identity associated with differentiable mappings. The identity is then used to derive the estimates of upper bound for mappings whose first derivatives absolute values are p-convex mappings. Four examples are also provided to illustrate the obtained results.

In this paper, the new version of the celebrated Montgomery identity is determined via quantum integral operators. By using it, certain quantum integral inequalities of Ostrowski type are established. Moreover, the relevant connection of the obtained results of this work with the derived results in previously published works is discussed.

Based on the established integral identity, some bounds involving the weighted Simpson type inequality are obtained where the first derivative of considered mappings is (m, h)-preinvex, boundedness or Lipschitzian. As applications, certain generalized inequalities in connection with weighted Simpson type quadrature formula, continuous random variables and F-divergence measures are investigated, respectively.

In this article, we introduce the notion of $\mathscr{M}$-convex functions, $\log$-$\mathscr{M}$-convex functions and the notion of quasi $\mathscr{M}$-convex functions. We derive some new analogues of Hermite-Hadamard like inequalities associated with $\mathscr{M}$-convex functions by using the concepts of ordinary, fractional and quantum calculus. The main results of this paper may be useful where bounds for natural phenomena described by integrals such as mechanical work are frequently required. These results are also helpful in the field of numerical analysis where error analysis is required.

We use the definition of a fractional integral operators, proposed by Katugampola, to establish a fractional Hermite-Hadamard’s inequality for p-convex mappings and an identity with two parameters. We derive several parameterized integral inequalities associated with this identity, and provide three examples to illustrate the obtained results.

We establish a (p, q)-integral identity with parameters and certain new (p, q)-integral inequalities of different types through (p, q)-differentiable mappings. Many results obtained in this article provide significant extensions of other related results given in the literature. Furthermore, we construct three examples to illustrate the investigated results.

We establish a Simpson-type identity of multiparameter and certain Simpson-type inequalities via k-fractional integrals. Worth mentioning, the obtained inequalities in this article generalize some results presented by Set et al. [Simpson type integral inequalities for convex functions via Riemann-Liouville integrals. Filomat. 2017;31(14):4415–4420] and Sarikaya et al. [On new inequalities of Simpson's type for s-convex functions. Comput Math Appl. 2010;60:2191–2199]. As applications, we also provide several inequalities for f-divergence measures and probability density functions. We expect that this study will be result in the new k-fractional integration explorations for Simpson-type inequalities.

A $k$ -fractional integral identity with multiple parameters is investigated. Based on this identity, some estimation-type results related to $k$ -fractional integral inequalities for the first-order differentiable functions are obtained. These results are then applied to the estimation of cumulative distribution function and some other special means.

Abstract In this paper, a new general identity for differentiable mappings via k-fractional integrals is derived. By using the concept of ( h , m ) $(h,m)$ -convexity, ( α , m ) $(\alpha,m)$ -convexity and the obtained equation, some new trapezium-like integral inequalities are established. The results presented provide extensions of those given in earlier works.

Abstract The authors first introduce the concepts of generalized ( α , m ) $(\alpha,m)$ -preinvex function, generalized quasi m-preinvex function and explicitly ( α , m ) $(\alpha, m)$ -preinvex function, and then provide some interesting properties for the newly introduced functions. The more important point is that we give a necessary and sufficient condition respecting the relationship between the generalized ( α , m ) $(\alpha, m)$ -preinvex function and the generalized quasi m-preinvex function. Second, a new Riemann-Liouville fractional integral identity involving twice differentiable function on m-invex is found. By using this identity, we establish the right-sided new Hermite-Hadamard-type inequalities via Riemann-Liouville fractional integrals for generalized ( α , m ) $(\alpha,m)$ -preinvex mappings. These inequalities can be viewed as generalization of several previously known results.

By using power-mean integral inequality and Hölder’s integral inequality, this paper establishes some new inequalities of Simpson type for functions whose three derivatives in absolute value are the class of (α, m)-geometric-arithmetically-convex functions. Finally, some applications to special means of positive real numbers have also been presented.

In this paper, we introduce and investigate generalized fractional integral operators containing the new generalized Mittag−Leffler function of two variables. We establish several new refinements of Hermite−Hadamard-like inequalities via co-ordinated convex functions.

Noticing that E -convexity, m-convexity and b-invexity have similar structures in their definitions, there are some possibilities to treat these three class of mappings uniformly. For this purpose, the definitions of the ( E , m ) -convex sets and the b- ( E , m ) -convex mappings are introduced. The properties concerning operations that preserve the ( E , m ) -convexity of the proposed mappings are derived. The unconstrained and inequality constrained b- ( E , m ) -convex programming are considered, where the sufficient conditions of optimality are developed and the uniqueness of the solution to the b- ( E , m ) -convex programming are investigated. Furthermore, the sufficient optimality conditions and the Fritz⁻John necessary optimality criteria for nonlinear multi-objective b- ( E , m ) -convex programming are established. The Wolfe-type symmetric duality theorems under the b- ( E , m ) -convexity, including weak and strong symmetric duality theorems, are also presented. Finally, we construct two examples in detail to show how the obtained results can be used in b- ( E , m ) -convex programming.

This paper concentrates on addressing fractional inequalities for exponential type convex functions. By means of exponential-type convexity, we firstly establish Hermite-Hadamard (HH) type inequalities for fractional integrals with exponential kernels. Secondly, based on the discovered fractional identity by separating [a; b] to n equal subintervals, and the fact that the twice derivative in absolute value is exponential type convex, we present multipoint-based HH inequalities, which cover the trapezoid- and Bullen-type inequalities for n = 1 and 2, correspondingly. During the period, some numerical examples with graphs are provided to show the validity of the deduced inequalities. [ABSTRACT FROM AUTHOR]

To investigate integral inclusions through interval valued mappings, we utilize the notion of the interval-valued fractional integrals having exponential kernels. We develop the reverse Minkowsky’s, including with the reverse Holder’s ¨ fractional integral inclusions in association with interval-valued mappings. Other fractional integral inclusions in relation to the Hermite–Hadamard-type inequalities are reported as well. Moreover, we provide some examples concerning interval valued mappings, to enucleate the correctness of the inclusion relations presented here. [ABSTRACT FROM AUTHOR]

In virtue of the conception of the multiplicative tempered fractional integrals, put forward by Fu et al. in the published article [AIMS Math., 6 (7): 7456-7478, 2021], we present a fractional integral identity for * differentiable functions. Based upon it, we develop several inequalities of Hermite-Hadamard type in association with * differentiable multiplicative mpreinvexity and (s,m) -preinvexity. [ABSTRACT FROM AUTHOR]

As an interesting generalization involving the interval-valued convex functions, the interval-valued exponential trigonometric convex function is firstly introduced, and their meaningful properties are then investigated. Meanwhile, certain Hermite-Hadamard- and Pachpattetype integral inclusion relations are also developed via the newly proposed functions in intervalvalued fractional calculus. In particular, an improved version of the Hermite-Hadamard's integral inclusions pertaining to the interval-valued exponential trigonometric convex functions is proposed as well. To identify the correctness of the derived inclusion relations in the study, the graphical representations for the outcomes are provided in terms of the change of the parameter α. [ABSTRACT FROM AUTHOR]