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Integral inequalities involving exponential convexity are significant in both theoretical and applied mathematics. In this paper, we establish a new Hermite-Hadamard type inequality for the class of exponentially convex functions by using the concept of $ (\alpha-s) $ exponentially convex function. Additionally, using the well-known Hermite-Hadamard and Ostrowski inequalities, we establish several new integral inequalities. These newly obtained results contain several well-known results as special cases. Finally, new estimations for the trapezoidal formula have been provided, illustrating the practical applications of the research.

Various differential and integral operators have been introduced and applied for the generalization of several integral inequalities. The purpose of this article is to create a more generalized fractional integral operator of Saigo type. This operator will be used alongwith the existing Saigo type and Q-Saigo type fractional integral operators to establish extended and generalized versions of several inequalities, including Pólya-Szegő and Chebyshev type inequalities.

This paper introduced and investigated a new form of convex mapping known as $ \alpha $-exponential type convexity. We presented several algebraic properties associated with this newly introduced convexity. Additionally, we established novel adaptations of well-known inequalities, including the Hermite-Hadamard and Ostrowski-type inequalities, specifically for this convex function. We also derived special cases of these newly established results. Furthermore, we provided new estimations for the trapezoidal formula, demonstrating practical applications of this research.

The well-known Hermite-Hadamard inequality for convex functions is extensively studied for different kinds of integrals and derivatives. This paper investigates some of its variants for $ q-h $-integrals using properties of convex functions. Inequalities for $ q $-integrals that have been published in recent years can be extracted from the main results of this paper.

This paper aims to unify q-derivative/q-integrals and h-derivative/h-integrals into a single definition, called q−h-derivative/q−h-integral. These notions are further extended on the finite interval [a,b] in the form of left and right q−h-derivatives and q−h-integrals. Some inequalities for q−h-integrals are studied and directly connected with well known results in diverse fields of science and engineering. The theory based on q-derivatives/q-integrals and h-derivatives/h-integrals can be unified using the q−h-derivative/q−h-integral concept.

In this paper, firstly, Hermite-Hadamard inequality via s-convex functions in the second sense using Caputo-Fabrizio fractional integral operator is established. We also compare our results with the existing ones. It is also shown that the obtained results are a generalization of the existing results. Finally, we give their applications to special means.

In this paper we establish some new inequalities of Ostrowski type for exponentially s-convex functions and s-convex functions on time scale. We also make comparison of our new results with already existing results by imposing some conditions.

This article aims to investigate unified versions of the well-known Hermite–Hadamard inequality by considering q-h-integrals and properties of convex functions. Currently published results for q-integrals can be deduced from inequalities of this paper. Moreover, some new results are presented in terms of corollaries.

Abstract In this paper, we establish Jensen’s inequality for s-convex functions in the first sense. By using Jensen’s inequalities, we obtain some Cauchy type means for p-convex and s-convex functions in the first sense. Also, by using Hermite–Hadamard inequalities for the respective generalized convex functions, we find new generalized Cauchy type means.

Abstract At first, we recall the q-operators in the context of q-calculus and by examining these operators we will introduce new definitions of the partial q-operators. Then, we investigate some new refinements inequalities of Hermite–Hadamard ( H − H $H-H$ ) type on the coordinated convex functions involving the new defined partial q-operators. From our main results, we establish several specific inequalities and we point out the existing results which had already been obtained in the literature.

This paper aims to establish generalized fractional integral inequalities for operators containing Mittag–Leffler functions. By applying (α,h−m)−p-convexity of real valued functions, generalizations of many well-known inequalities are obtained. Hadamard-type inequalities for various classes of functions are given in particular cases.

In this paper, we propose a new family of interval valued ($\mathrm{IV}$) convex functions termed as generalized modified $(p,h)$-convex $\mathrm{IV}$ functions. We obtain the counterpart of Hermite-Hadamard $H\cdot H$ inequality by extending the $\mathrm{IV}$ fractional integral to the $\mathrm{IV}$ $\psi_{k}$-Riemann-Liouville ($\psi_{k}-RL$) fractional integrals. Also, several inequalities using extended operations on the newly defined class of convex $\mathrm{IV}$ functions are given.

Abstract In this paper, we obtain the Hermite–Hadamard and Hermite–Hadamard–Fejer type inequalities for p-convex functions via conformable fractional integrals. We also discuss some special cases.

Abstract In this research article, we establish some Hermite–Hadamard type inequalities for tgs $tgs$-convex functions via Katugampola fractional integrals and ψ-Riemann–Liouville fractional integrals. Through these results we give some new Hermite–Hadamard type inequalities for tgs $tgs$-convex functions via Riemann–Liouville fractional integrals and classical integrals.

In this paper, we obtain ostrowski type inequalities for exponentially convex function and exponentially s-convex function in second sense. Applications to some special means are also obtain. Here we extend the results of some previous investigations.

In this paper, we aim to find unified estimates of fractional integrals involving Mittag–Leffler functions in kernels. The results obtained in terms of fractional integral inequalities are provided for various kinds of convex and related functions. A variant of Hadamard-type inequality is also presented, which shows the upper and lower bounds of fractional integral operators of many kinds. The results of this paper are directly linked with many recently published inequalities.

This paper investigates several integral inequalities held simultaneously for q and h-integrals in implicit form. These inequalities are established for symmetric functions using certain types of convex functions. Under certain conditions, Hadamard-type inequalities are deducible for q-integrals. All the results are applicable for ℏ-convex, m-convex and convex functions defined on the non-negative part of the real line.

Convex bodies are symmetric in nature. Between the two variables of symmetry and convexity, a correlation connection is also perceptible. Due to the interchangeable analogous properties, the application on either of them has been practicable in these modern years. The current analysis sheds insight on a general new identity involving a number of parameters for a twice partial quantum differentiable function. We find several unique quantum integral inequalities by using the new identity and a twice partial quantum differentiable function whose absolute value is coordinated convex. In addition, we present several novel and interesting error estimation-like results related to the well-known quantum Hermite–Hadamard inequality. Some examples are provided at the end to support and demonstrate the effectiveness of the new outcomes.

Abstract This research explores Caputo k-fractional integral inequalities for functions whose nth order derivatives are absolutely continuous and possess Grüss type variable bounds. Using Chebyshev inequality (Waheed et al. in IEEE Access 7:32137–32145, 2019) for Caputo k-fractional derivatives, several integral inequalities are derived. Further, Laplace transform of Caputo k-fractional derivative is presented and Caputo k-fractional derivative and Riemann–Liouville k-fractional integral of an extended generalized Mittag-Leffler function are calculated. Moreover, using the extended generalized Mittag-Leffler function, Caputo k-fractional differential equations are presented and their solutions are proposed by applying the Laplace transform technique.

In this paper, we establish some Hermite-Hadamard type inequalities via $\psi$-Riemann-Liouville fractional integrals for $s$-convex functions in second sense and the functions belongs to the class $P(I)$ $($that is, a class of non-negative functions $\curlyvee:I\rightarrow\mathbb{R}$ which satisfies the condition $\curlyvee(ra_1+(1-r)a_2)\leq \curlyvee(a_1)+\curlyvee(a_2)$, for all $a_1,a_2\in I$ and $r\in[0,1])$. Some applications to special means are also investigated.

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 give some correct quantum type Simpson’s inequalities via the application of q-Hölder’s inequality. The inequalities of this study are compatible with famous Simpson’s 1/8 and 3/8 quadrature rules for four and six panels, respectively. Several special cases from our results are discussed in detail. A counter example is presented to explain the limitation of Hölder’s inequality in the quantum framework.

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.

We consider Jensen’s functionals on time scales and discuss its properties and applications. Further, we define weighted generalized and power means on time scales. By applying the properties of Jensen’s functionals on these means, we obtain several refinements and converses of Hölder’s inequality on time scales.

We apply the theory of isotonic linear functionals to derive a series of known inequalities, extensions of known inequalities, and new inequalities in the theory of dynamic equations on time scales.

We discuss log-convexity for the differences of the Popoviciu inequalities and introduce some mean value theorems and related results. Also we give the Cauchy means of the Popoviciu type and we show that these means are monotonic.

We will introduce new means of Cauchy's type Mr,ls(f,μ) defined, for example, as Mr,ls(f,μ)=((l(l−s)/r(r−s))(Mrr(f,μ)−Msr(f,μ)/Mll(f,μ)−Msl(f,μ)))1/(r−l), in the case when l ≠ r ≠ s,  l,r ≠ 0. We will show that this new Cauchy's mean is monotonic, that is, the following result. Theorem. Let t,r,u,v∈â„Â, such that t≤v, r≤u. Then for Mr,ls(f,μ), one has Mt,rs≤Mv,us. We will also give some related comparison results.

We give an improvement and a reversion of the well-known Ky-Fan inequality as well as some related results.