Some Hadamard–Fejér Type Inequalities for LR-Convex Interval-Valued Functions

The purpose of this study is to introduce the new class of Hermite–Hadamard inequality for LR-convex interval-valued functions known as LR-interval Hermite–Hadamard inequality, by means of pseudo-order relation ( ≤p ). This order relation is defined on interval space. We have proved that if the interval-valued function is LR-convex then the inclusion relation “ ⊆ ” coincident to pseudo-order relation “ ≤p ” under some suitable conditions. Moreover, the interval Hermite–Hadamard–Fejér inequality is also derived for LR-convex interval-valued functions. These inequalities also generalize some new and known results. Useful examples that verify the applicability of the theory developed in this study are presented. The concepts and techniques of this paper may be a starting point for further research in this area.

Some Inequalities for LR-$$\left({h}_{1}, {h}_{2}\right)$$-Convex Interval-Valued Functions by Means of Pseudo Order Relation

In both theoretical and applied mathematics fields, integral inequalities play a critical role. Due to the behavior of the definition of convexity, both concepts convexity and integral inequality depend on each other. Therefore, the relationship between convexity and symmetry is strong. Whichever one we work on, we introduced the new class of generalized convex function is known as LR-$$\left({h}_{1}, {h}_{2}\right)$$ h 1 , h 2 -convex interval-valued function (LR-$$\left({h}_{1}, {h}_{2}\right)$$ h 1 , h 2 -IVF) by means of pseudo order relation. Then, we established its strong relationship between Hermite–Hadamard inequality (HH-inequality)) and their variant forms. Besides, we derive the Hermite–Hadamard–Fejér inequality (HH–Fejér inequality)) for LR-$$\left({h}_{1}, {h}_{2}\right)$$ h 1 , h 2 -convex interval-valued functions. Several exceptional cases are also obtained which can be viewed as its applications of this new concept of convexity. Useful examples are given that verify the validity of the theory established in this research. This paper’s concepts and techniques may be the starting point for further research in this field.

Some generalized Hermite–Hadamard–Fejér inequality for convex functions

In this paper, we have established some generalized inequalities of Hermite–Hadamard–Fejér type for generalized integrals. The results obtained are applied for fractional integrals of various type and therefore contain some previous results reported in the literature.

Refinements of the integral Jensen’s inequality generated by finite or infinite permutations

There are a lot of papers dealing with applications of the so-called cyclic refinement of the discrete Jensen’s inequality. A significant generalization of the cyclic refinement, based on combinatorial considerations, has recently been discovered by the author. In the present paper we give the integral versions of these results. On the one hand, a new method to refine the integral Jensen’s inequality is developed. On the other hand, the result contains some recent refinements of the integral Jensen’s inequality as elementary cases. Finally some applications to the Fejér inequality (especially the Hermite–Hadamard inequality), quasi-arithmetic means, and f-divergences are presented.

New fuzzy-interval inequalities in fuzzy-interval fractional calculus by means of fuzzy order relation

<abstract> <p>It is well-known that interval analysis provides tools to deal with data uncertainty. In general, interval analysis is typically used to deal with the models whose data are composed of inaccuracies that may occur from certain kinds of measurements. In interval analysis and fuzzy-interval analysis, the inclusion relation (⊆) and fuzzy order relation $\left(\preccurlyeq \right)$ both are two different concepts, respectively. In this article, with the help of fuzzy order relation, we introduce fractional Hermite-Hadamard inequality (<italic>HH</italic>-inequality) for <italic>h</italic>-convex fuzzy-interval-valued functions (<italic>h</italic>-convex-IVFs). Moreover, we also establish a strong relationship between <italic>h</italic>-convex fuzzy-IVFs and Hermite-Hadamard Fejér inequality (<italic>HH</italic>-Fejér inequality) via fuzzy Riemann Liouville fractional integral operator. It is also shown that our results include a wide class of new and known inequalities for <italic>h</italic>-convex fuzz-IVFs and their variant forms as special cases. Nontrivial examples are presented to illustrate the validity of the concept suggested in this review. This paper's techniques and approaches may serve as a springboard for further research in this field.</p> </abstract>

New Hermite-Hadamard inequalities in fuzzy-interval fractional calculus via exponentially convex fuzzy interval-valued function

<abstract><p>In the present note, we develop Hermite-Hadamard type inequality and He's inequality for exponential type convex fuzzy interval-valued functions via fuzzy Riemann-Liouville fractional integral and fuzzy He's fractional integral. Moreover, we establish Hermite-Fejér inequality via fuzzy Riemann-Liouville fractional integral.</p></abstract>

Refinement of fejér inequality for convex and co-ordinated convex functions

In this paper, we shall establish the co-ordinated convex function. It can connect to the right-hand side of Fejér inequality in two variables and thus a new refinement can be found. In addition, some applications to estimates for Euler’s Beta function are also given in the end.

Hermite-Hadamard-Fejér Type Inequalities for Preinvex Functions Using Fractional Integrals

In this paper, we have established the Hermite–Hadamard–Fejér inequality for fractional integrals involving preinvex functions. The results presented here provide new extensions of those given in earlier works as the weighted estimates of the left and right hand side of the Hermite–Hadamard inequalities for fractional integrals involving preinvex functions doesn’t exist previously.

Trapezoidal Type Fejér Inequalities Related to Harmonically Convex Functions and Application

Some authors introduced the concepts of the harmonically arithmetic convex functions and establish some integral inequalities of Hermite Hadamard Fejér type related to the harmonically arithmetic convex functions. In this paper, a mapping M(t) is considered to get some preliminary results and a new trapezoidal form of Fejér inequality related to the harmonically arithmetic convex functions. By using a mapping M(t), the new theorems and corollaries are obtained. Taking advantage of these, applications were given for some real number averages.