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In this paper, we are concerned with the study of the existence and uniqueness of fixed points for the class of functions f : C → C satisfying the inequality ℓ (α f (t) + (1 − α) f (s)) ≤ σ ℓ (α t + (1 − α) s) for every t , s ∈ C with f (t) ≠ f (s) , where C is a closed subset of [ 0 , ∞) , α , σ ∈ (0 , 1) are constants, and ℓ : [ 0 , ∞) → [ 0 , ∞) is a function satisfying the condition inf t > 0 ℓ (t) t ρ > 0 for some constant ρ > 0. Namely, under a weak continuity condition imposed on f , we show that f possesses a unique fixed point, and for every t 0 ∈ C , the Picard sequence defined by t n + 1 = f (t n) , n ≥ 0 , converges to this fixed point. Next, we study the special cases when C is a closed interval and ℓ is a convex or concave function. Namely, making use of the Hermite-Hadamard inequalities, we obtain several new fixed point theorems. To the best of our knowledge, the considered class of functions was never previously investigated in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the study of the existence and uniqueness of fixed points for the class of functions $ f: C\to C $ satisfying the inequality$ \ell\left(\alpha f(t)+(1-\alpha)f(s)\right)\leq \sigma \ell(\alpha t+(1-\alpha)s) $for every $ t, s\in C $ with $ f(t)\neq f(s) $, where $ C $ is a closed subset of $ [0, \infty) $, $ \alpha, \sigma\in (0, 1) $ are constants, and $ \ell: [0, \infty)\to [0, \infty) $ is a function satisfying the condition $ \inf_{t > 0} \frac{\ell(t)}{t^\rho} > 0 $ for some constant $ \rho > 0 $. Namely, under a weak continuity condition imposed on $ f $, we show that $ f $ possesses a unique fixed point, and for every $ t_0\in C $, the Picard sequence defined by $ t_{n+1} = f(t_n) $, $ n\geq 0 $, converges to this fixed point. Next, we study the special cases when $ C $ is a closed interval and $ \ell $ is a convex or concave function. Namely, making use of the Hermite-Hadamard inequalities, we obtain several new fixed point theorems. To the best of our knowledge, the considered class of functions was never previously investigated in the literature.

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]

The purpose of this study is to establish recent inequalities based on double integrals of mappings whose higher-order partial derivatives in absolute value are convex on the co-ordinates on rectangle from the plane. Also, some special cases of results improved in this study are examined. [ABSTRACT FROM AUTHOR]

The main aim this work is to give Hermite–Hadamard inequalities for convex functions via generalized proportional fractional integrals. We also obtained extensions of Hermite–Hadamard type inequalities for generalized proportional fractional integrals. [ABSTRACT FROM AUTHOR]

In current study, we establish some new quantum Hermite Hadamard-Jensen-Mercer type integral inequalities by way of recently newly defined integral.Then we investigate the connections between our results and those in earlier works.Furthermore, we present some examples which satisfied our main outcomes. We expect that this innovative way opens many avenues for interested researchers will reconcile discovering further quantum approximations of Hermite-Hadamard type variants for other classes of convex functions, and, additionally, to discover uses in the aforementioned scientific disciplines. [ABSTRACT FROM AUTHOR]

We first construct new Hermite-Hadamard type inequalities which include generalized fractional integrals for convex functions by using an operator which generates some significant fractional integrals such as Riemann-Liouville fractional and the Hadamard fractional integrals. Afterwards, Trapezoid and Midpoint type results involving generalized fractional integrals for functions whose the derivatives in modulus and their certain powers are convex are established. We also recapture the previous results in the particular situations of the inequalities which are given in the earlier works.

Abstract 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.

This work establishes some new inequalities to find error bounds for Maclaurin's formulas in the framework of q-calculus. For this, we first prove an integral identity involving q-integral and q-derivative. Then, we use this new identity to prove some q-integral inequalities for q-differentiable convex functions. The inequalities proved here are very important in the literature because, with their help, we can find error bounds for Maclaurin's formula in both q and classical calculus. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The class of symmetric function interacts extensively with other types of functions. One of these is the class of convex functions, which is closely related to the theory of symmetry. In this paper, we obtain some new fractional Hermite–Hadamard inequalities with an exponential kernel for subadditive functions and for their product, and some known results are recaptured. Moreover, using a new identity as an auxiliary result, we deduce several inequalities for subadditive functions pertaining to the new fractional integrals involving an exponential kernel. To validate the accuracy of our results, we offer some examples for suitable choices of subadditive functions and their graphical representations. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some Hermite-Hadamard type inequalities for uniformly convex functions with respect to geodesic in Hadamard space. Also, we give some application of these functions in means. [ABSTRACT FROM AUTHOR]

In this work, Hermite-Hadamard type inequalities for subadditive functions via quantum integrals are established. Moreover, Hermite-Hadamard type inequalities for the product of two subadditive functions are also obtained. It is worth to mentioning that some existing inequalities of Hermite-Hadamard type for subadditive functions are obtained by considering the limit of the real number q ∈ (0,1) as q → 1− in the key results. [ABSTRACT FROM AUTHOR]

A review on Hermite-Hadamard type inequalities connected with a different classes of convexities and fractional differential operators is presented. In the various classes of convexities it includes, classical convex functions, quasi-convex functions, p-convex functions, strongly-m-convex functions, strongly-(θ,m)-convex functions, (s,m)-convex functions, (θ, h - m)-convex functions, strongly (θ, h-m)-convex functions, (h,m)-convex functions of the second type, m-convex functions, h-convex functions, (h,m)-convex functions, relative-convex functions, exponentially (θ, h - m)- convex functions, harmonically h-convex functions and geometric-arithmetically s-convex functions. In the fractional differential operators it includes, Caputo fractional derivative, k-Caputo fractional derivative and Hilfer fractional derivative. [ABSTRACT FROM AUTHOR]

In this work, Hölder-Isc, an inequality is used for the class of n-times differentiable (s;m)- convex functions. The outcomes are new Hermite-Hadamard type inequalities and modified integrals are estimated by better bounds. Special cases are deduced as the existing results from literature. Furthermore, some applications to arithmetic, geometric and logarithmic means are also presented. [ABSTRACT FROM AUTHOR]

The goal of this study is to create new variations of the well-known Hermite–Hadamard inequality (HH-inequality) for preinvex interval-valued functions (preinvex I-V-Fs). We develop several additional inequalities for the class of functions whose product is preinvex I-V-Fs. The findings described here would be generalizations of those found in previous studies. Finally, we obtain the Hermite–Hadamard–Fejér inequality with the support of preinvex interval-valued functions. Some new and classical special cases are also obtained. Moreover, some nontrivial examples are given to check the validity of our main results. [ABSTRACT FROM AUTHOR]

This work establishes some new inequalities to find error bounds for Maclaurin’s formulas in the framework of q-calculus. For this, we first prove an integral identity involving q-integral and q-derivative. Then, we use this new identity to prove some q-integral inequalities for q-differentiable convex functions. The inequalities proved here are very important in the literature because, with their help, we can find error bounds for Maclaurin’s formula in both q and classical calculus.

A new generalization of the Katugampola generalized fractional integrals in terms of the Mittag-Leffler functions is proposed. Consequently, new generalizations of the Hermite-Hadamard inequalities by this newly proposed fractional integral operator, for a positive convex stochastic process, are established. Other known results are easily deduced as particular cases of these inequalities. The obtained results also hold for any convex function. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Hermite-Hadamard inequality for the Riemann-Liouville integral transformations for uniformly convex functions from a geodesic perspective in the Hadamard spaces. This inequality is widely used in some fractional integral approximations. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new versions of Hermite–Hadamard type inequalities for co-ordinated convex functions via q 1 , q 2 -integrals. Since the inequalities are newly proved, we therefore consider some examples of co-ordinated convex functions and show their validity for particular choices of q 1 , q 2 ∈ (0 , 1) . We hope that the readers show their interest in these results. [ABSTRACT FROM AUTHOR]

The class of symmetric function interacts extensively with other types of functions. One of these is the class of convex functions, which is closely related to the theory of symmetry. In this paper, we obtain some new fractional Hermite–Hadamard inequalities with an exponential kernel for subadditive functions and for their product, and some known results are recaptured. Moreover, using a new identity as an auxiliary result, we deduce several inequalities for subadditive functions pertaining to the new fractional integrals involving an exponential kernel. To validate the accuracy of our results, we offer some examples for suitable choices of subadditive functions and their graphical representations.

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. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain Fejér type inequalities for higher order convex functions and the general weighted integral formula involving w-harmonic sequences of functions. Also, Fejér type inequalities are obtained for the general three, four and five point quadrature formulae. Further, Fejér type inequalities for the corrected three, corrected four and corrected five point quadrature formulae are considered. In special cases, Fejér type estimates for Simpson, Maclaurin, corrected Simpson and corrected Maclaurin quadrature rules are derived. [ABSTRACT FROM AUTHOR]

The connection between generalized convexity and symmetry has been studied by many authors in recent years. Due to this strong connection, generalized convexity and symmetry have arisen as a new topic in the subject of inequalities. In this paper, we introduce the concept of interval-valued preinvex functions on the coordinates in a rectangle from the plane and prove Hermite–Hadamard type inclusions for interval-valued preinvex functions on coordinates. Further, we establish Hermite–Hadamard type inclusions for the product of two interval-valued coordinated preinvex functions. These results are motivated by the symmetric results obtained in the recent article by Kara et al. in 2021 on weighted Hermite–Hadamard type inclusions for products of coordinated convex interval-valued functions. Our established results generalize and extend some recent results obtained in the existing literature. Moreover, we provide suitable examples in the support of our theoretical results. [ABSTRACT FROM AUTHOR]

In the present study, two new classes of convex functions are established with the aid of Raina’s function, which is known as the ψ-s-convex and ψ-quasi-convex functions. As a result, some refinements of the Hermite–Hadamard (ℋℋ{\mathcal{ {\mathcal H} {\mathcal H} }})-type inequalities regarding our proposed technique are derived via generalized ψ-quasi-convex and generalized ψ-s-convex functions. Considering an identity, several new inequalities connected to the ℋℋ{\mathcal{ {\mathcal H} {\mathcal H} }} type for twice differentiable functions for the aforesaid classes are derived. The consequences elaborated here, being very broad, are figured out to be dedicated to recapturing some known results. Appropriate links of the numerous outcomes apprehended here with those connecting comparatively with classical quasi-convex functions are also specified. Finally, the proposed study also allows the description of a process analogous to the initial and final condition description used by quantum mechanics and special relativity theory.

Abstract In this paper, certain Hermite–Jensen–Mercer type inequalities are proved via conformable integrals of arbitrary order. We establish some different and new fractional Hermite–Hadamard–Mercer type inequalities for a differentiable function f whose derivatives in the absolute values are convex.

Abstract In this paper, certain Hermite–Hadamard–Mercer type inequalities are proved via k-Caputo fractional derivatives. We established some new k-Caputo fractional derivatives inequalities with Hermite–Hadamard–Mercer type inequalities for differentiable mapping ψ ( n ) $\psi^{(n)}$ whose derivatives in the absolute values are convex.

The present paper first establishes that an identity involving generalized fractional integrals is proved for differentiable functions by using two parameters. By utilizing this identity, we obtain several parameterized inequalities for the functions whose derivatives in absolute value are convex. Finally, we show that our main inequalities reduce to Ostrowski type inequalities, Simpson type inequalities and trapezoid type inequalities which are proved in earlier published papers. [ABSTRACT FROM AUTHOR]

In this paper, we tend to establish some new Hermite–Hadamard type integral inequalities for multiplicatively convex function on coordinates and for product of two multiplicatively convex functions on coordinates. [ABSTRACT FROM AUTHOR]

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]

In the paper, the authors establish some new Hermite–Hadamard type inequalities for harmonically convex functions via generalized fractional integrals. Moreover, the authors prove extensions of the Hermite–Hadamard inequality for harmonically convex functions via generalized fractional integrals without using the harmonic convexity property for the functions. The results offered here are the refinements of the existing results for harmonically convex functions. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we introduce the notion of exponentially p-convex function and exponentially s-convex function in the second sense. We establish several Hermite–Hadamard type inequalities for exponentially p-convex functions and exponentially s-convex functions in second sense. The present investigation is an extension of several well known results.

In this paper, we establish some new versions of Hermite–Hadamard type inequalities for co-ordinated convex functions via q1,q2-integrals. Since the inequalities are newly proved, we therefore consider some examples of co-ordinated convex functions and show their validity for particular choices of q1,q2∈(0,1). We hope that the readers show their interest in these results.

The main objective of this paper is to establish some new Hermite–Hadamard type inequalities involving k-Riemann–Liouville fractional integrals. Using the convexity of differentiable functions some related inequalities have been proved, which have deep connection with some known results. At the end, some applications of the obtained results in error estimations of quadrature formulas are also considered. [ABSTRACT FROM AUTHOR]

In this paper we present a very short proof of inequalities of Hermite-Hadamard type obtained by M. Bessenyei and Zs. Páles. This proof is based on the recently developed method connected with use of stochastic orderings of random variables. In the second part of the paper we present a way to extend these known inequalities. Namely, we describe completely the possible inequalities of Hermite-Hadamard type for longer expression than it was the case in the results of Bessenyei and Páles. [ABSTRACT FROM AUTHOR]

In this paper, we establish Hermite-Hadamard type inequalities for uniformly p-convex functions. Also, a new fractional Hermite-Hadamard type inequality for convex functions is obtained by using only the left Riemann-Liouville fractional integral. Finally some estimation of left fractional integration studies for Hermite-Hadamard type inequalities. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we obtain the Hermite–Hadamard type inequalities for s-convex functions and m-convex functions via a generalized fractional integral, known as Katugampola fractional integral, which is the generalization of Riemann–Liouville fractional integral and Hadamard fractional integral. We show that through the Katugampola fractional integral we can find a Hermite–Hadamard inequality via the Riemann–Liouville fractional integral.

The main ob jective of this paper is to study the classical Hermite- Hadamard-type inequalities for Riemann-liouville fractional integrals. The Hermite-Hadamard inequalities are derived for k -Riemann-Liouville fractional integrals by using the definitions of different types of convex functions such as s-convex function, m-convex function, (s, m)- convex function and twice differentiable geometric-arithmetically s-convex function. Our results generalize the results of [8] and [18]. [ABSTRACT FROM AUTHOR]

In this paper, certain Hermite–Hadamard–Mercer type inequalities are proved via k-Caputo fractional derivatives. We established some new k-Caputo fractional derivatives inequalities with Hermite–Hadamard–Mercer type inequalities for differentiable mapping ψ (n) whose derivatives in the absolute values are convex. [ABSTRACT FROM AUTHOR]

In this study, we first establish some integral inequalities of Hermite-Hadamard type in the setting of multiplicative calculus for multiplicatively P-functions. Then, by using some properties of this kind of functions, we obtain new inequalities involving multiplicative integrals for product and quotient of multiplicatively P-functions and convex functions. [ABSTRACT FROM AUTHOR]

In this paper, we establish inequalities of Hermite–Hadamard type for harmonically convex functions using a generalized fractional integral. The results of our paper are an extension of previously obtained results (İşcan in Hacet. J. Math. Stat. 43(6):935–942, 2014 and İşcan and Wu in Appl. Math. Comput. 238:237–244, 2014). We also discuss some special cases for our main results and obtain new inequalities of Hermite–Hadamard type. [ABSTRACT FROM AUTHOR]

In this paper, we give new definitons related to Katugampola fractional integral for two variables functions. We are interested in giving the Hermite{Hadamard inequality for a rectangle in plane via convex functions on co-ordinates involving Katugampola fractional integral. [ABSTRACT FROM AUTHOR]

Let s 0 be a fixed point of a strictly increasing continuous function β. Hamza et al. introduced the quantum operator D β [ g ] (θ) : = g (β (θ)) − g (θ) β (θ) − θ , θ ≠ s 0 and D β [ g ] (s 0) : = g ′ (s 0) if θ = s 0. For specific choices of the function β one obtains the known Jackson q -operator D q as well as the Hahn quantum operator D q , ω. Regarding its inverse operator, the β -integral, we establish the corresponding β -Hermite-Hadamard inequalities. Among others, we also obtain the Hermite-Hadamard type inequalities for the Jackson q -integral, the Nörlund integral and for the Jackson-Thomae (or Jackson-Nörlund) q , ω -integral. • The main result is Theorem 3.1: it establishes the Hermite-Hadamard inequalities for the β -integral. • Subsection 3.3 comprises, as corollaries, the Hermite-Hadamard inequalities for all known quantum integrals. • Section 4 contains an application to special means and also a statistic interpretation. [ABSTRACT FROM AUTHOR]

In this paper, firstly, Hermite–Hadamard–Fejér type inequalities for p-convex functions are built. Secondly, an integral identity and some Hermite–Hadamard–Fejér type integral inequalities for p-convex functions are obtained. Finally, some Hermite–Hadamard and Hermite–Hadamard–Fejér inequalities for convex, harmonically convex and p-convex functions are given. Some results presented here for p-convex functions provide extensions of others given in earlier works for convex and harmonically convex and p-convex functions.

Since 2004 more than 80 articles were published related to superquadracity. Here we put together some of these results, in particular those related to Jensen's inequality in one and in several variables, to Jensen-Steffensen inequalities, to Hardy's inequalities and to Hermite-Hadamard and Fejer inequalities. These inequalities are analogs of inequalities satisfied by convex functions and in those cases that the superquadratic functions are non-negative we get refinements of inequalities satisfied by convex functions. [ABSTRACT FROM AUTHOR]

In this paper, with a new approach, a new fractional Hermite-Hadamard type inequalities for convex functions is obtained by using only the left Riemann-Liouville fractional integral. Also, to have new fractional trapezoid and midpoint type inequalities for the differentiable convex functions, two new equalities are proved. Our results generalize earlier studies. We expect that this study will be lead to the new fractional integration studies for Hermite-Hadamard type inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we prove three new Katugampola fractional Hermite-Hadamard type inequalities for harmonically convex functions by using the left and the right fractional integrals independently. One of our Katugampola fractional Hermite-Hadamard type inequalities is better than given in [17]. Also, we give two new Katugampola fractional identities for differentiable functions. By using these identities, we obtain some new trapezoidal type inequalities for harmonically convex functions. Our results generalize many results from earlier papers. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of convex functions which is called strongly (p; h)-convex functions. We show that this class includes several other new classes of convex functions. We also establish some new results for Hermite-Hadamard type inequalities via strongly (p; h)-convex functions. Some special cases which can be deduced from our main results are also discussed. [ABSTRACT FROM AUTHOR]