Search

In this paper, we prove some new inequalities of Hermite–Hadamard type for differentiable functions with h-convex derivatives. It is also shown that the newly established inequalities are extension of the existing inequalities in the literature. Finally, we give applications of the new results and outline some future problems. [ABSTRACT FROM AUTHOR]

The significance of fractional calculus cannot be underestimated, as it plays a crucial role in the theory of inequalities. In this paper, we study a new class of mean-type inequalities by incorporating Riemann-type fractional integrals. By doing so, we discover a novel set of such inequalities and analyze them using different mathematical identities. This particular class of inequalities is introduced by employing a generalized convexity concept. To validate our work, we create visual graphs and a table of values using specific functions to represent the inequalities. This approach allows us to demonstrate the validity of our findings and further solidify our conclusions. Moreover, we find that some previously published results emerge as special consequences of our main findings. This research serves as a catalyst for future investigations, encouraging researchers to explore more comprehensive outcomes by using generalized fractional operators and expanding the concept of convexity. [ABSTRACT FROM AUTHOR]

In this paper, we establish some inequalities involving both h 1 -convex functions and h 2 -concave functions such that (h 1 , h 2) is an increasing pair of functions. We apply the obtained results to Breckner's s-convex functions, P-convex functions and Godunova–Levin functions. In order to unify the results, we introduce generalized s-convex functions of the second kind. [ABSTRACT FROM AUTHOR]

In this paper, we state some characterizations of h-convex function defined on a convex set in a linear space. By doing so, we extend the Jensen–Mercer inequality for h-convex function. We present the concept of operator h-convex functions and give some operator versions of Jensen and Jensen–Mercer type inequalities for some classes of operator h-convex functions and unital positive linear maps. Finally, we introduce the complementary inequality of Jensen's inequality for h-convex functions. [ABSTRACT FROM AUTHOR]

In this study, the Hermite–Hadamard–Fejér inequalities for GA-h-convex are proved, and the results for particular classes of functions are highlighted. In addition, several generalizations of the Hermite–Hadamard inequalities are presented. Some features of functions H and F that are naturally linked to the Hermite–Hadamard–Fejér-type inequalities for GA-h-convex have also been discussed. Finally, we obtain applications of the results related to the p-logarithmic mean and the mean of order p.

The significance of fractional calculus cannot be underestimated, as it plays a crucial role in the theory of inequalities. In this paper, we study a new class of mean-type inequalities by incorporating Riemann-type fractional integrals. By doing so, we discover a novel set of such inequalities and analyze them using different mathematical identities. This particular class of inequalities is introduced by employing a generalized convexity concept. To validate our work, we create visual graphs and a table of values using specific functions to represent the inequalities. This approach allows us to demonstrate the validity of our findings and further solidify our conclusions. Moreover, we find that some previously published results emerge as special consequences of our main findings. This research serves as a catalyst for future investigations, encouraging researchers to explore more comprehensive outcomes by using generalized fractional operators and expanding the concept of convexity.

In this work, we generalize and extend the concept of superquadratic functions by introducing h-superquadratic functions. Some basic properties of h-superquadraticity are elaborated and investigated. A connection between h-convexity and h-superquadraticity is provided. Inequalities of Jensen’s and Jensen–Mercer’s type with their converses are proved. [ABSTRACT FROM AUTHOR]

In this paper, we prove Jensen's operator inequality for an h-convex function and we point out the results for classes of continuous fields of operators. Also, some generalizations of Jensen's operator inequality and some properties of the h-convex function are given. [ABSTRACT FROM AUTHOR]

In this paper, the authors established a new identity for differentiable functions, afterward, they obtained some new general inequalities for functions whose first derivatives in absolute value at certain powers are h-convex by using the identity. On the other hand, a general inequality is studied, which gives Hermite-Hadamard, Bullen and Simpson inequalities. Also, they gave some applications for special means for arbitrary positive numbers. [ABSTRACT FROM AUTHOR]

We obtain some new inequalities of Hermite–Hadamard type. We consider functions that have convex or generalized convex derivative. Additional inequalities are proven for functions whose second derivative in absolute values are convex. Applications of the main results are presented. [ABSTRACT FROM AUTHOR]

Abstract We derive some properties and results for a new extended class of convex functions, generalized strongly modified h-convex functions. Moreover, we discuss Schur-type, Hermite–Hadamard-type, and Fejér-type inequalities for this class. The crucial fact is that this extended class has awesome properties similar to those of convex functions.

In this paper we investigate the property of engulfing for H-convex functions defined on the Heisenberg group H n . Starting from the horizontal sections introduced by Capogna and Maldonado (Proc Am Math Soc 134:3191–3199, 2006) , we consider a new notion of section, called H n -section, as well as a new condition of engulfing associated to the H n -sections, for an H-convex function defined in H n. These sections, that arise as suitable unions of horizontal sections, are dimensionally larger; as a matter of fact, the H n -sections, with their engulfing property, will lead to the definition of a quasi-distance in H n in a way similar to Aimar et al. in the Euclidean case (J Fourier Anal Appl 4:377–381, 1998). A key role is played by the property of round H-sections for an H-convex function, and by its connection with the engulfing properties. [ABSTRACT FROM AUTHOR]

The authors have tried to prove some Ostrowski-type fractional integral inequalities for r¡times differentiable functions and generalized some results which were carried out in both [6, 25]. Some applications to special means are given. The methods and techniques of this paper could further stimulate the research conducted in the field of fractional integral inequalities. [ABSTRACT FROM AUTHOR]

A characterization of h-convex function via Hermite-Hadamard inequality related to the h-convex functions is investigated. In fact it is determined that under what conditions a function is h-convex, if it satisfies the h-convex version of Hermite-Hadamard inequality.

In this paper, we use two new fractional integral operators with exponential kernel about the midpoint of the interval to construct some Hermite–Hadamard type fractional integral inequalities for h-convex functions. Taking two integral identities about the first and second derivatives of the function as auxiliary functions, the main results are obtained by using the properties of h-convexity and the module. In order to illustrate the application of the results, we propose four examples and plot function images to intuitively present the meaning of the inequalities in the main results, and we verify the correctness of the conclusion. This study further expands the generalization of Hermite–Hadamard-type inequalities and provides some research references for the study of Hermite–Hadamard-type inequalities.

In this paper, the authors established a new identity for differentiable functions, afterwards they obtained some new inequalities for functions whose first derivatives in absolute value at certain powers are h-convex by using the identity. Also they give some applications for special means for arbitrary positive numbers. [ABSTRACT FROM AUTHOR]

In this work, Popoviciu type inequalities for h-MN-convex functions are proved. Some direct examples are pointed out. [ABSTRACT FROM AUTHOR]

In this article, the author established a new identity for differentiable functions and obtained some new inequalities for differentiable functions based on h-convex functions including Bullen-type inequalities. Also, some applications are given special means for arbitrary positive numbers. [ABSTRACT FROM AUTHOR]

Integral inequalities involving many fractional integral operators are used to solve various fractional differential equations. In the present paper, we will generalize the Hermite–Jensen–Mercer-type inequalities for an h-convex function via a Caputo–Fabrizio fractional integral. We develop some novel Caputo–Fabrizio fractional integral inequalities. We also present Caputo–Fabrizio fractional integral identities for differentiable mapping, and these will be used to give estimates for some fractional Hermite–Jensen–Mercer-type inequalities. Some familiar results are recaptured as special cases of our results.

The Hermite–Hadamard inequality was first considered for convex functions and has been studied extensively. Recently, many extensions were given with the use of general convex functions. In this paper we present some variants of the Hermite–Hadamard inequality for general convex functions in the context of q-calculus. From our theorems, we deduce some recent results in the topic. [ABSTRACT FROM AUTHOR]

In this paper, we establish new Ostrowski type inequalities involving fractional integrals with respect to another function. Such fractional integrals generalize the Riemann-Liouville fractional integrals and the Ostrowski type fractional integrals. [ABSTRACT FROM AUTHOR]

We derive some properties and results for a new extended class of convex functions, generalized strongly modified h-convex functions. Moreover, we discuss Schur-type, Hermite–Hadamard-type, and Fejér-type inequalities for this class. The crucial fact is that this extended class has awesome properties similar to those of convex functions. [ABSTRACT FROM AUTHOR]

In this work, we introduce the class of h-MN-convex functions by generalizing the concept of MN-convexity and combining it with h-convexity. Namely, Let I, J be two intervals subset of (0, ∞) such that (0, 1) ⊆ J and [a, b] ⊆ I. Consider a non-negative function h : (0, ∞) → (0, ∞) and let M : [0, 1] → [a, b] (0 < a < b) be a Mean function given by M(t) = M(h(t); a, b); where by M(h(t); a, b) we mean one of the following functions: Ah(a, b) ≔ h(1 – t) a + h(t) b, Gh(a, b) = ah(1–t)bh(t) and ; with the property that M(h(0); a, b) and M(h(1); a, b) = b. A function f : I → (0, ∞) is said to be h-MN-convex (concave) if the inequality holds for all x, y ∈ I and t ∈ [0, 1], where M and N are two mean functions. In this way, nine classes of h-MN-convex functions are established and some of their analytic properties are explored and investigated. Characterizations of each type are given. Various Jensen's type inequalities and their converses are proved. [ABSTRACT FROM AUTHOR]

The theory of fractional analysis has been a focal point of fascination for scientists in mathematical science, given its essential definitions, properties, and applications in handling real-life problems. In the last few decades, many mathematicians have shown their considerable interest in the theory of fractional calculus and convexity due to their wide range of applications in almost all branches of applied sciences, especially in numerical analysis, physics, and engineering. The objective of this article is to establish Hermite-Hadamard type integral inequalities by employing the k-Riemann-Liouville fractional operator and its refinements, whose absolute values are twice-differentiable h-convex functions. Moreover, we also present some special cases of our presented results for different types of convexities. Moreover, we also study how q-digamma functions can be applied to address the newly investigated results. Mathematical integral inequalities of this class and the arrangements associated have applications in diverse domains in which symmetry presents a salient role.

The principal motivation of this paper is to establish a new integral equality related to k-Riemann Liouville fractional operator. Employing this equality, we present several new inequalities for twice differentiable convex functions that are associated with Hermite–Hadamard integral inequality. Additionally, some novel cases of the established results for different kinds of convex functions are derived. This fractional integral sums up Riemann–Liouville and Hermite–Hadamard’s inequality, which have a symmetric property. Scientific inequalities of this nature and, particularly, the methods included have applications in different fields in which symmetry plays a notable role. Finally, applications of q-digamma and q-polygamma special functions are presented.

In this paper, a condition which implies the continuity of an h -convex function is investigated. In fact, any h -convex function bounded from above is continuous if the function h satisfies a certain condition which is called pre-continuity condition. [ABSTRACT FROM AUTHOR]

Abstract Some basic results in connection with the area balance function associated to a Lebesgue integrable function are obtained and then two new Fejér trapezoidal type inequalities are presented. Also some applications for random variable, trapezoidal formula and special means are given. [ABSTRACT FROM AUTHOR]

In this paper, we give some characterizations of h-convex functions, and some applications related to these functions are also obtained. According to these results, we can know better the relation between convex functions and h-convex functions. [ABSTRACT FROM AUTHOR]

In this paper we derive some new quantum estimates of Fejér type inequalities which involve Riemann type of quantum integrals via some classes of convex functions. We also discuss some special cases which can be deduced from the main results of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are h-convex and h-concave by using power mean inequality, Hölder inequality and some other integral inequalities. Some applications to special means of real numbers are also given. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we will point out a similar inequality to Hadamard’s for h-convex function defined on a disk. Some mappings connected with this inequality and related results are also obtained. [Copyright &y& Elsevier]

In this paper, we derive new integral inequalities for functions with h-convex and h-concave first derivatives. As a consequence, we give new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are h-convex and h-concave and we point out the results for some special classes of functions. [ABSTRACT FROM AUTHOR]

In this paper, we establish some Simpson type inequalities for functions whose third derivatives in the absolute value are h-convex and (α, m)--convex, respectively. [ABSTRACT FROM AUTHOR]

In this paper, some Ostrowski type inequalities via Riemann-Liouville fractional integrals for h-convex functions, which are super-multiplicative or super-additive, are given. These results not only generalize those of [24, 25], but also provide new estimates on these types of Ostrowski inequalities for fractional integrals. [ABSTRACT FROM AUTHOR]

In the paper, the authors introduce a notion 'log - h-convex functions' and establish several Hermite-Hadamard type integral inequalities for this kind of functions. [ABSTRACT FROM AUTHOR]