Preinvex Functions Research Articles

Some new (p,q)‐Hadamard‐type integral inequalities for the generalized m‐preinvex functions

This paper introduces and establishes new Hermite‐Hadamard type inequalities specifically tailored for ‐preinvex functions within the ‐calculus framework. These newly developed inequalities come with accompanying left‐right estimates, which enhance their practical utility. The primary objective of this research is to investigate the properties of ‐differentiable ‐preinvex functions and derive inequalities that extend and generalize existing results in the domain of integral inequalities. The techniques employed in this study hold broader implications, finding relevance in various fields where symmetry is paramount. The findings presented in this paper make a significant contribution to the field of analytic inequalities, offering valuable insights into the behavior and characteristics of ‐preinvex functions. Moreover, the established results demonstrate the wider applicability and generalization of analogous findings from prior literature. The techniques and inequalities introduced herein pave the way for further exploration and research in the realm of integral inequalities.

W-Invexity and optimality problem

As the development of a family of preinvex functions, we define the w-invex set and a class of real functions. These are functions with the names w-preinvex, w-strictly preinvex, w-prequasi-invex, w-strictly prequasi-invex, w-semi-strictly prequasi-invex, and w-pre-pseudo-invex. The fundamental characteristics of these functions are thoroughly examined. Several examples are provided here that serve to clarify the concept in this context. Several of these preinvex functions are addressed to investigate the optimization problems further.

Generalized integral Jensen inequality

In this paper we introduce necessary and sufficient conditions for a real-valued function to be preinvex. Some properties of preinvex functions and new versions of Jensen’s integral type inequality in this setting are given. Several examples are also involved.

Weighted Fejér, Hermite–Hadamard, and Trapezium-Type Inequalities for (h1,h2)–Godunova–Levin Preinvex Function with Applications and Two Open Problems

This note introduces a new class of preinvexity called (h1,h2)-Godunova-Levin preinvex functions that generalize earlier findings. Based on these notions, we developed Hermite-Hadamard, weighted Fejér, and trapezium type inequalities. Furthermore, we constructed some non-trivial examples in order to verify all the developed results. In addition, we discussed some applications related to the trapezoidal formula, probability density functions, special functions and special means. Lastly, we discussed the importance of order relations and left two open problems for future research. As an additional benefit, we believe that the present work can provide a strong catalyst for enhancing similar existing literature.

Hermite-Hadamard, Fejér and trapezoid type inequalities using Godunova-Levin Preinvex functions via Bhunia's order and with applications to quadrature formula and random variable.

Convex and preinvex functions are two different concepts. Specifically, preinvex functions are generalizations of convex functions. We created some intriguing examples to demonstrate how these classes differ from one another. We showed that Godunova-Levin invex sets are always convex but the converse is not always true. In this note, we present a new class of preinvex functions called $ (\mathtt{h_1}, \mathtt{h_2}) $-Godunova-Levin preinvex functions, which is extensions of $ \mathtt{h} $-Godunova-Levin preinvex functions defined by Adem Kilicman. By using these notions, we initially developed Hermite-Hadamard and Fejér type results. Next, we used trapezoid type results to connect our inequality to the well-known numerical quadrature trapezoidal type formula for finding error bounds by limiting to standard order relations. Additionally, we use the probability density function to relate trapezoid type results for random variable error bounds. In addition to these developed results, several non-trivial examples have been provided as proofs.

ON GENERAL LOCAL FRACTIONAL INTEGRAL INEQUALITIES FOR GENERALIZED H-PREINVEX FUNCTIONS ON YANG’S FRACTAL SETS

In this paper, based on Yang’s fractal theory, the Hermite–Hadamard’s inequalities for generalized [Formula: see text]-preinvex function are proved. Then, using the local fractional integral identity proposed by Sun [Some local fractional integral inequalities for generalized preinvex functions and applications to numerical quadrature, Fractals 27(5) (2019) 1950071] as auxiliary function, some parameterized local fractional integral inequalities for generalized [Formula: see text]-preinvex functions are established. For the special cases of the parameters, some generalized Simpson-type, midpoint-type and trapezoidal inequalities are established. Finally, some applications of these inequalities in numerical integration are proposed.

Inequalities for Hyperbolic Type Harmonic Preinvex Function

In the present paper, we have introduced a new class of preinvexity namely hyperbolic type harmonic preinvex functions and to support this new definition, some of its algebraic properties are elaborated. By using this new class of preinvexity, we have established a few Hermite-Hadamard type integral inequalities. Some novel refinements of Hemite-Hadamard type inequalities for hyperbolic type harmonic preinvex functions are presented as well. Finally, the Riemann-Liouville fractional version of the Hermite-Hadamard Inequality is established.

Ostrowski-Type Fractional Integral Inequalities: A Survey

This paper presents an extensive review of some recent results on fractional Ostrowski-type inequalities associated with a variety of convexities and different kinds of fractional integrals. We have taken into account the classical convex functions, quasi-convex functions, (ζ,m)-convex functions, s-convex functions, (s,r)-convex functions, strongly convex functions, harmonically convex functions, h-convex functions, Godunova-Levin-convex functions, MT-convex functions, P-convex functions, m-convex functions, (s,m)-convex functions, exponentially s-convex functions, (β,m)-convex functions, exponential-convex functions, ζ¯,β,γ,δ-convex functions, quasi-geometrically convex functions, s−e-convex functions and n-polynomial exponentially s-convex functions. Riemann–Liouville fractional integral, Katugampola fractional integral, k-Riemann–Liouville, Riemann–Liouville fractional integrals with respect to another function, Hadamard fractional integral, fractional integrals with exponential kernel and Atagana-Baleanu fractional integrals are included. Results for Ostrowski-Mercer-type inequalities, Ostrowski-type inequalities for preinvex functions, Ostrowski-type inequalities for Quantum-Calculus and Ostrowski-type inequalities of tensorial type are also presented.

Certain generalized Riemann–Liouville fractional integrals inequalities based on exponentially (h,m)-preinvexity

In this article, we define a new preinvex mapping known as the exponentially (h,m)-preinvex. We explore the properties of this mapping, and derive several Φk-Riemann–Liouville fractional integrals inequalities through exponentially (h,m)-preinvexity. Using these results, we present certain estimates for Hermite–Hadamard-type inequalities, and also deduce several inequalities for the fractional integrals of k-Riemann–Liouville, which are special cases of the main results. During this period, we give several numerical examples to further demonstrate the correctness of our main results. As applications, we propose a few inequalities according to special means.

Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals

Abstract In this article, we have established some new bounds of Fejér-type Hermite-Hadamard inequality for k k -fractional integrals involving r r -times differentiable preinvex functions. It is noteworthy that in the past, there was no weighted version of the left and right sides of the Hermite-Hadamard inequality for k k -fractional integrals for generalized convex functions available in the literature.

Hermite–Hadamard-type Inequalities for hbar-preinvex Interval-Valued Functions via Fractional Integral

We present a comprehensive study on Hermite–Hadamard-type inequalities for interval-valued functions that are hbar-preinvex, using the Riemann–Liouville fractional integral. Our research extends and generalizes some existing results found in the literature. In addition, we provide accurate proofs for the main theorems originally derived by Srivastava et al. in their publication titled “Hermite–Hadamard Type Inequalities for Interval-Valued Preinvex Functions via Fractional Integral Operators" (Int. J. Comput. Int. Sys. 15(1):8, 2022). Finally, we illustrate our findings through a practical example to demonstrate the validity of our results.

A Comprehensive Review of the Hermite–Hadamard Inequality Pertaining to Quantum Calculus

A review of results on Hermite–Hadamard (H-H) type inequalities in quantum calculus, associated with a variety of classes of convexities, is presented. In the various classes of convexities this includes classical convex functions, quasi-convex functions, p-convex functions, (p,s)-convex functions, modified (p,s)-convex functions, (p,h)-convex functions, tgs-convex functions, η-quasi-convex functions, ϕ-convex functions, (α,m)-convex functions, ϕ-quasi-convex functions, and coordinated convex functions. Quantum H-H type inequalities via preinvex functions and Green functions are also presented. Finally, H-H type inequalities for (p,q)-calculus, h-calculus, and (q−h)-calculus are also included.

On local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) \left({\tilde{h}}_{1},{\tilde{h}}_{2}) -preinvexity involving local fractional integral operators with Mittag-Leffler kernel

Abstract Local fractional integral inequalities of Hermite-Hadamard type involving local fractional integral operators with Mittag-Leffler kernel have been previously studied for generalized convexities and preinvexities. In this article, we analyze Hermite-Hadamard-type local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) \left({\tilde{h}}_{1},{\tilde{h}}_{2}) -preinvex function comprising local fractional integral operators and Mittag-Leffler kernel. In addition, two examples are discussed to ensure that the derived consequences are correct. As an application, we construct an inequality to establish central moments of a random variable.

Some New Fractional Hadamard and Pachpatte-Type Inequalities with Applications via Generalized Preinvexity

The term convexity associated with the theory of inequality in the sense of fractional analysis has a broad range of different and remarkable applications in the domain of applied sciences. The prime objective of this article is to investigate some new variants of Hermite–Hadamard and Pachpatte-type integral inequalities involving the idea of the preinvex function in the frame of a fractional integral operator, namely the Caputo–Fabrizio fractional operator. By employing our approach, a new fractional integral identity that correlates with preinvex functions for first-order differentiable mappings is presented. Moreover, we derive some refinements of the Hermite–Hadamard-type inequality for mappings, whose first-order derivatives are generalized preinvex functions in the Caputo–Fabrizio fractional sense. From an application viewpoint, to represent the usability of the concerning results, we presented several inequalities by using special means of real numbers. Integral inequalities in association with convexity in the frame of fractional calculus have a strong relationship with symmetry. Our investigation provides a better image of convex analysis in the frame of fractional calculus.

Generalized AB-Fractional Operator Inclusions of Hermite–Hadamard’s Type via Fractional Integration

The aim of this research is to explore fractional integral inequalities that involve interval-valued preinvex functions. Initially, a new set of fractional operators is introduced that uses the extended generalized Mittag-Leffler function Eμ,α,lγ,δ,k,c(τ;p) as a kernel in the interval domain. Additionally, a new form of Atangana–Baleanu operator is defined using the same kernel, which unifies multiple existing integral operators. By varying the parameters in Eμ,α,lγ,δ,k,c(τ;p), several new fractional operators are obtained. This study then utilizes the generalized AB integral operators and the preinvex interval-valued property of functions to establish new Hermite–Hadamard, Pachapatte, and Hermite–Hadamard–Fejer inequalities. The results are supported by numerical examples, graphical illustrations, and special cases.

Pre-Invexity and Fuzzy Fractional Integral Inequalities via Fuzzy Up and Down Relation

The symmetric function class interacts heavily with other types of functions. One of these is the pre-invex function class, which is strongly related to symmetry theory. This paper proposes a novel fuzzy fractional extension of the Hermite-Hadamard, Hermite-Hadamard-Fejér, and Pachpatte type inequalities for up and down pre-invex fuzzy-number-valued mappings. Using fuzzy fractional operators, several generalizations have been developed, where well-known results fit as particular cases. Additionally, some non-trivial examples are included to support the discussion and the applicability of the key findings. The approach appears trustworthy and effective for dealing with various nonlinear problems in science and engineering. The findings are general and may constitute contributions to complex waveform theory.

Generalized m -preinvexity on fractal set and related local fractional integral inequalities with applications

In this work, we address and explore the concept of generalized \(m\)-preinvex functions on fractal sets along with linked local fractional integral inequalities. Additionally, some engrossing algebraic properties are presented to facilitate the current initiated idea. Furthermore, we prove the latest variant of Hermite-Hadamard type inequality employing the proposed definition of preinvexity. We also derive several novel versions of inequalities of the Hermite-Hadamard type and Fejér-Hermite-Hadamard type for the first-order local differentiable generalized \(m\)-preinvex functions. Finally, some new inequalities for the generalized means and generalized random variables are established as applications.

Some novel refinements of Hermite-Hadamard and Pachpatte type integral inequalities involving a generalized preinvex function pertaining to Caputo-Fabrizio fractional integral operator

<abstract><p>In this article, we aim to introduce and explore a new class of preinvex functions called $ \mathfrak{n} $-polynomial $ m $-preinvex functions, while also presenting algebraic properties to enhance their numerical significance. We investigate novel variations of Pachpatte and Hermite-Hadamard integral inequalities pertaining to the concept of preinvex functions within the framework of the Caputo-Fabrizio fractional integral operator. By utilizing this direction, we establish a novel fractional integral identity that relates to preinvex functions for differentiable mappings of first-order. Furthermore, we derive some novel refinements for Hermite-Hadamard type inequalities for functions whose first-order derivatives are polynomial preinvex in the Caputo-Fabrizio fractional sense. To demonstrate the practical utility of our findings, we present several inequalities using specific real number means. Overall, our investigation sheds light on convex analysis within the context of fractional calculus.</p></abstract>

On generalizations of some integral inequalities for preinvex functions via (p,q)-calculus

In this paper, we establish some new (p,q)-integral inequalities of Simpson’s second type for preinvex functions. Many results given in this paper provide generalizations and extensions of the results given in previous research. Moreover, some examples are given to illustrate the investigated results.

Quantum ostrowski type inequalities for pre-invex functions

Abstract In this paper, using the quantum derivatives and quantum integrals, we prove some new quantum Ostrowski’s type inequalities for pre-invex functions. Furthermore, in the special cases of newly developed inequalities, we obtain different new and existing Ostrowski’s type inequalities.