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In the context of quantum symmetric calculus, this study proposed more refined version of Ostrowski and Hermite–Hadamard type inequalities. The function involved in these inequalities are convex functions. In order to reach the target, left and right quantum symmetric derivative and corresponding integral are used. Furthermore, the Hölder inequality is established in the frame work of left and right quantum symmetric integral. The new results refined the results about integral inequalities that exist in the literature. [ABSTRACT FROM AUTHOR]

Integral inequalities and the Mittag-Leffler function play a crucial role in many branches of mathematics and applications, including fractional calculus, mathematical physics, and engineering. In this paper, we introduced an extended generalized Mittag-Leffler function that involved several well-known Mittag-Leffler functions as a special case. We also introduced an associated generalized fractional integral to obtain some estimates for fractional integral inequalities of the Hermite-Hadamard and Hermite-Hadamard-Fejér types. This article offered several analytical tools that will be useful to anyone working in this field. To demonstrate the veracity of our findings, we offered a few numerical and graphical examples. A few applications of modified Bessel functions and unitarily invariant norm of matrices were also given.

This paper presents enhanced parameterized quantum Hermite-Hadamard type integral inequalities for functions whose third right and left q-derivatives in absolute value are strongly convex functions. We obtain new bounds using H o ¨ lder’s and power mean inequalities as primary tools. Also, we derive new quantum estimates for q-trapezoidal and q-midpoints type inequalities in specific scenarios, which we illustrate with examples. These outcomes possess the potential for practical applications in optimizing various economic problems. [ABSTRACT FROM AUTHOR]

In this article, we investigate a new class of convexity named inverse cosine convex functions (ICCF). We explore and examine some algebraic and geometric features by presenting the graphs of several significant ICCF functions for visual demonstration. By using this proposed class of functions, we derive the Hermite–Hadamard (HH) inequality and certain refinements applicable to functions whose first derivative in absolute value is an ICCF. To derive the main consequences, we mainly use Hölder's, Hölder–İşcan, and power‐mean integral inequalities. Furthermore, it is proved that the results obtained using Hölder–İşcan and improved power‐mean integral inequalities have better approximations compared to other techniques. This novel idea of inverse cosine convexity allows us to explore new horizons of research and may motivate researchers to investigate such classes of convexity. [ABSTRACT FROM AUTHOR]

In this article, we adopt the tempered fractional integral operators to develop some novel Minkowski and Hermite-Hadamard type integral inequalities. Thus, we give several special cases of the integral inequalities for tempered fractional integrals obtained in the earlier works. [ABSTRACT FROM AUTHOR]

The main motivation in this article is to prove new integral identities and related results. In this paper, we deal with E`-convex function, Hermite-Hadamard type inequalities, and Katugampola fractional integrals. Firstly, we established the Hermite-Hadamard type inequalities for E`-convex function via Katugampola fractional integrals operators. After that we obtain several new identities and give new results of Holder's inequality, Jensen's integral inequality, Young's inequality, and Power mean inequality for E`-convex function by using Katugampola fractional integral operators. We make connections of our results with various results recognized in the literature.

This article explores the integration of HermiteHadamard Type Inequalities and convex functions within the domain of signal processing, elucidating their theoretical underpinnings and practical implications. Beginning with a comprehensive background, we focus on the historical context and foundational concepts that underlie these mathematical constructs. Our discussion progresses to articulate the problem formulation, delineating the specific challenges and objectives addressed in the study. The theoretical framework elucidates the HermiteHadamard Type Inequalities, highlighting their mathematical formulations, properties, and fundamental proofs. Concurrently, the discourse unfolds the theory and properties of convex functions, elucidating their significance and applications within signal processing paradigms. With a focus on applications, we illustrate the utility of Hermite-Hadamard Type Inequalities and convex functions in signal processing tasks. Through empirical studies and case examples, we demonstrate their efficacy in signal denoising, compression, and feature extraction, showcasing tangible results and comparative analyses. We discuss the challenges and limitations inherent in the application of these mathematical constructs in real-world scenarios, thereby paving the way for future research directions and advancements. Finally, we conclude by summarizing the key insights gleaned from our exploration and underscore the profound implications of Hermite-Hadamard Type Inequalities and convex functions in shaping the landscape of contemporary signal processing methodologies.

Novel Hermite–Hadamard type inequalities for p-convex functions utilizing the Caputo–Fabrizio fractional integral are established. Additionally, fresh inequalities incorporating the Caputo–Fabrizio fractional integral operator are examined. Furthermore, applications to special means of the primary outcomes are presented. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to present new q-parametrized Hermite–Hadamard-like type integral inequalities for functions whose third quantum derivatives in absolute values are s-convex and  (α , m) -convex, respectively. Two new q-integral identities are presented for three time q-differentiable functions. These lemmas are used like basic elements in our proofs, along with several important tools like q-power mean inequality, and q-Holder's inequality. In a special case, a non-trivial example is considered for a specific parameter and this case illustrates the investigated results. We make links between these findings and several previous discoveries from the literature. [ABSTRACT FROM AUTHOR]

Our paper explores Hermite–Hadamard inequalities through the application of Abel–Gontscharoff Green's function methodology, which involves interpolating polynomials and Riemann-type generalized fractional integrals. While establishing our main results, we explore new identities. These identities are used to estimate novel findings for functions, such that the second derivative of the functions is monotone, absolutely convex, and concave. A section relating the results of exploration to generalized means and trapezoid formulas is included in the applications. We anticipate that the method presented in this study will inspire further research in this field. [ABSTRACT FROM AUTHOR]

The main goal of this work is to provide quantum parametrized Hermite-Hadamard like type integral inequalities for functions whose second quantum derivatives in absolute values follow different type of convexities. A new quantum integral identity is derived for twice quantum differentiable functions, which is used as a key element in our demonstrations along with several basic inequalities such as: power mean inequality, and Holder's inequality. The symmetry of the Hermite-Hadamard type inequalities is stressed by the different types of convexities. Several special cases of the parameter are chosen to illustrate the investigated results. Four examples are presented. [ABSTRACT FROM AUTHOR]

In this paper, with the use of newly defined class up and down log–convex fuzzy-number valued mappings, we offer a few new and original mappings defined by applying some mild restrictions over the definition of up and down log–convex fuzzy-number valued mapping. With the use of these mappings, we are able to develop partners of Fejér-type inequalities for up and down log–convexity, which improve upon certain previously established findings. The discussion also includes these mappings' characteristics. Moreover, some nontrivial examples are also provided to prove the validation of our main results. [ABSTRACT FROM AUTHOR]

There is significant interaction between the class of symmetric functions and other types of functions. The multiplicative convex function class, which is intimately related to the idea of symmetry, is one of them. In this paper, we obtain some new generalized multiplicative fractional Hermite–Hadamard type inequalities for multiplicative convex functions and for their product. Additionally, we derive a number of inequalities for multiplicative convex functions related to generalized multiplicative fractional integrals utilising a novel identity as an auxiliary result. We provide some examples for the appropriate selections of multiplicative convex functions and their graphical representations to verify the authenticity of our main results. [ABSTRACT FROM AUTHOR]

Convex bodies are naturally symmetrical. There is also a correlation between the two variables of symmetry and convexity. Their use, in either case, has been feasible in recent years because of their interchangeable and similar properties. The proposed analysis provides information on a new class for a convex function which is known as up and down X 1 , X 2 -convex fuzzy-Number valued mappings ( U D - X 1 , X 2 -convex F N V M ). Using this class, we disclosed a number of new versions of integral inequalities. Additionally, we give a number of new related integral inequalities connected to the well-known Hermite-Hadamard-type inequalities. In conclusion, some examples are given to back up and show the value of these new results. [ABSTRACT FROM AUTHOR]

Integral inequalities play a crucial role in both theoretical and applied mathematics. Because of the relevance of these notions, we have discussed a new class of introduced generalized convex function called as coordinated left and right convex interval-valued function (coordinated LR-convex IVF) using the pseudo-order relation (≤p). On interval space, this order relation is defined. First, a pseudo-order relation is used to show Hermite-Hadamard type inequality (HH type inequality) for coordinated LR-convex IVF. Second for coordinated LR-convex IVF, Some HH type inequalities are also derived for the product of two coordinated LR-convex IVFs. Furthermore, we have demonstrated that our conclusions cover a broad range of new and well-known inequalities for coordinated LR-convex IVFs and their variant forms as special instances which are defined by Zhao et al. and Budak et al. Finally, we have shown that the inclusion relation "⊇" confidents to the pseudo-order relation "≤p" for coordinated LR-convex IVFs. The concepts and methodologies presented in this study might serve as a springboard for additional research in this field, as well as a tool for investigating probability and optimization research, among other things.

This study investigates the Iqb-differentiability and Iqb -integrability for interval-valued functions defined on the q-geometric set. We also establish some Iqb -Hermite-Hadamard type inequalities. Furthermore, some examples are presented to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, the Hermite-Hadamard type inequalitis for Riemann-Liouville fractional integrals via strongly h-convex functions are established. Furthermore, we obtain some identities related to the fractional integrals with n-times differentiable functions, and then gain midpoint type and trapezoid type error estimates connected with the Hermite-Hadamard type inequalities, which generalize some known results. [ABSTRACT FROM AUTHOR]

We introduce and investigate the coordinated log-h-convexity for interval-valued functions. Also, we prove some new Jensen type inequalities and Hermite-Hadamard type inequalities, which generalize some known results in the literature. Moreover, some examples are given to illustrate our results.

We introduce the concepts of $\mathbb{K}$-Riemann integral and radial $\mathbb{K}$-g$H$-derivative for interval-valued functions. We also give some important properties of interval-valued $\mathbb{K}$-Riemann integral, and extend interval-valued Hermite-Hadamard type inequalities in the case of $\mathbb{K}$-Riemann integral. Several examples are shown to illustrate the results.

As is known, integral inequalities related to convexity have a close relationship with symmetry. In this paper, we introduce a new notion of interval-valued harmonically m , h 1 , h 2 -Godunova–Levin functions, and we establish some new Hermite–Hadamard inequalities. Moreover, we show how this new notion of interval-valued convexity has a close relationship with many existing definitions in the literature. As a result, our theory generalizes many published results. Several interesting examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]

Copyright of Erzincan University Journal of Science & Technology is the property of Erzincan Binali Yildirim Universitesi and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Journal of Zhejiang University (Science Edition) is the property of Journal of Zhejiang University (Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Abstract In this paper, we introduce the non-convex interval-valued functions for fuzzy-interval-valued functions, which are called -convex fuzzy-interval-valued functions, by means of fuzzy order relation. This fuzzy order relation is defined level-wise through Kulisch–Miranker order relation given on the interval space. By using the -convexity concept, we present fuzzy-interval Hermite–Hadamard inequalities for fuzzy-interval-valued functions. Several exceptional cases are debated, which can be viewed as useful applications. Interesting examples that verify the applicability of the theory developed in this study are presented. The results of this paper can be considered as extensions of previously established results.

Abstract In this paper we establish some new results on trapezium type inequalities of coordinated distance-disturbed ( ℓ 1 , h 1 ) $(\ell _{1},h_{1})$ – ( ℓ 2 , h 2 ) $(\ell _{2},h_{2})$ -convex functions of higher orders ( σ 1 , σ 2 ) $(\sigma _{1},\sigma _{2})$ by using the Katugampola ( k 1 , k 2 ) $(k_{1},k_{2})$ -fractional integrals. As special cases of our general results, we recapture some earlier proved results.

The purpose of this study is to define a new class of harmonically convex functions, which is known as left and right harmonically convex interval-valued function (LR--convex IV-F), and to establish novel inclusions for a newly defined class of interval-valued functions (IV-Fs) linked to Hermite–Hadamard (H-H) and Hermite–Hadamard–Fejér (H-H-Fejér) type inequalities via interval-valued Riemann–Liouville fractional integrals (IV-RL-fractional integrals). We also attain some related inequalities for the product of two LR--convex IV-Fs. These findings enable us to identify a new class of inclusions that may be seen as significant generalizations of results proved by Iscan and Chen. Some examples are included in our findings that may be used to determine the validity of the results. The findings in this work can be seen as a considerable advance over previously published findings. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we introduce the concepts of map T $\mathcal {T}$ and interval-valued T $\mathcal {T}$ -convex, and give some basic properties. Further, we extend fractional Hermite–Hadamard inequalities in the case of T $\mathcal {T}$ -convex and Ostrowski type inequalities for interval-valued functions. Several examples are presented to illustrate the results.

Abstract The concept of generalized h-preinvex function on real linear fractal sets R β $R^{\beta }$ ( 0 < β ≤ 1 $0 < \beta \le 1$ ) is introduced, which extends generalized preinvex, generalized s-preinvex, generalized Godunova–Levin preinvex, and generalized P-preinvex functions. In addition, some Hermite–Hadamard type inequalities for these classes of functions involving local fractional integrals are established. Lastly, the upper bounds for generalized expectation, generalized rth moment, and generalized variance of a continuous random variable are given to illustrate the applications of the obtained results.

Abstract In this paper, some new inequalities of the Hermite–Hadamard type for the classes of functions whose derivatives’ absolute values are m-convex and ( α , m ) $( \alpha , m) $ -convex are obtained. The results obtained in this work extend and improve the corresponding ones in the literature. Some applications to special means of real numbers are also given.

Abstract We use the definition of a new class of fractional integral operators, recently introduced by Ahmad et al. in [J. Comput. Appl. Math. 353:120–129, 2019], to establish a fractional-type integral identity with one parameter. We derive some parameterized integral inequalities for convex mappings based on this identity, and provide two examples to illustrate the investigated results as well. Moreover, we present applications of our findings to special means of real numbers, and error estimations for the quadrature formula in numerical analysis.

In this paper, we examine a generalized convexity type inequality, called e-convexity. [ABSTRACT FROM AUTHOR]

In this paper, by using two local fractional integral operators with Mittag-Leffler kernel, we construct some new Hermite–Hadamard type and Hermite–Hadamard–Fejér type local fractional integral inequalities for generalized preinvex functions on Yang's fractal sets. Finally, three examples are proposed to illustrate the main results. Meanwhile, we also proposed an identity and a bounded estimate for the moments of random variables to illustrate the applications of the results. [ABSTRACT FROM AUTHOR]

Abstract We introduce the concept of interval harmonically convex functions. By using two different classes of convexity, we get some further refinements for interval fractional Hermite–Hadamard type inequalities. Also, some examples are presented.

Abstract In this paper, we adopt conformable fractional integral to develop integral inequalities such as Minkowski and Hermite–Hadamard inequalities. Our results are the generalization of the inequalities obtained by Dahmani and Bougoffa cited in the literature.

In this paper, we show that harmonic convex functions $f$ is strongly $(p, h)$-harmonic convex functions if and only if it can be decomposed as $g(x) = f(x) - c (\frac{1}{x^p})^2,$ where $g(x)$ is $(p, h)$-harmonic convex function. We obtain some new estimates class of strongly $(p, h)$-harmonic convex functions involving hypergeometric and beta functions. As applications of our results, several important special cases are discussed. We also introduce a new class of harmonic convex functions, which is called strongly $(p, h)$-harmonic $\log$-convex functions. Some new Hermite-Hadamard type inequalities for strongly $(p, h)$-harmonic $log$-convex functions are obtained. These results can be viewed as important refinement and significant improvements of the new and previous known results. The ideas and techniques of this paper may stimulate further research.

In this paper, the definition of generalized s-preinvex function on Yang's fractal sets ℝ γ (0 < γ ≤ 1) is proposed, and the generalized Hermite–Hadamard's inequality for this class of functions is established. By using this convexity, some generalized Hermite–Hadamard type integral inequalities with parameters are established. For these inequalities, the absolute values of twice local fractional order derivative of the functions are generalized s-preinvex functions. Some special integral inequalities can be obtained by assigning special values to the obtained inequalities, and two examples are given to illustrate our results. Finally, we propose the applications of the results in numerical integration and error estimation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the non-convex interval-valued functions for fuzzy-interval-valued functions, which are called -convex fuzzy-interval-valued functions, by means of fuzzy order relation. This fuzzy order relation is defined level-wise through Kulisch–Miranker order relation given on the interval space. By using the -convexity concept, we present fuzzy-interval Hermite–Hadamard inequalities for fuzzy-interval-valued functions. Several exceptional cases are debated, which can be viewed as useful applications. Interesting examples that verify the applicability of the theory developed in this study are presented. The results of this paper can be considered as extensions of previously established results. [ABSTRACT FROM AUTHOR]

In this paper we establish some new results on trapezium type inequalities of coordinated distance-disturbed (ℓ 1 , h 1) – (ℓ 2 , h 2) -convex functions of higher orders (σ 1 , σ 2) by using the Katugampola (k 1 , k 2) -fractional integrals. As special cases of our general results, we recapture some earlier proved results. [ABSTRACT FROM AUTHOR]

In this paper, we establish some Hermite‐Hadamard type inequalities for the Generalized Riemann‐Liouville fractional integrals Ia+,gαf and Ib−,gαf, where g is a strictly increasing function on a,b, having a continuous derivative on a,b and under the assumption that the composite function f∘g−1 is h ‐convex on ga,gb. Some applications for Hadamard fractional integrals and s‐Godunova‐Levin type convex functions are also provided. [ABSTRACT FROM AUTHOR]

Some inequalities of Hermite-Hadamard type for GA-convex functions defined on positive intervals are given.

Since the so-called Hermite-Hadamard inequality for a convex function was presented, its extensions, refinements, and variants, which are called Hermite-Hadamard type inequalities, have been extensively investigated. In this paper, we aim to establish two Hermite-Hadamard type inequalities and an identity for convex functions associated with known fractional conformable integral operators. Also the results presented here are indicated to reduce to relatively simple known results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concepts of map T and interval-valued T -convex, and give some basic properties. Further, we extend fractional Hermite–Hadamard inequalities in the case of T -convex and Ostrowski type inequalities for interval-valued functions. Several examples are presented to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, some new inequalities of the Hermite–Hadamard type for the classes of functions whose derivatives' absolute values are m-convex and (α , m) -convex are obtained. The results obtained in this work extend and improve the corresponding ones in the literature. Some applications to special means of real numbers are also given. [ABSTRACT FROM AUTHOR]

We use the definition of a new class of fractional integral operators, recently introduced by Ahmad et al. in [J. Comput. Appl. Math. 353:120–129, 2019], to establish a fractional-type integral identity with one parameter. We derive some parameterized integral inequalities for convex mappings based on this identity, and provide two examples to illustrate the investigated results as well. Moreover, we present applications of our findings to special means of real numbers, and error estimations for the quadrature formula in numerical analysis. [ABSTRACT FROM AUTHOR]

Abstract We firstly give a modification of the known Hermite-Hadamard type inequalities for the generalized k-fractional integral operators of a function with respect to another function. We secondly establish several Hermite-Hadamard type inequalities for the generalized k-fractional integral operators of a function with respect to another function. The results presented here, being very general, are pointed out to be specialized to yield some known results. Relevant connections of the various results presented here with those involving relatively simple fractional integral operators are also indicated.

The main aim of this paper is to consider a new class of harmonic convex functions with respect to an arbitrary non-negative function, which is called strongly (p, h)-harmonic convex function. We establish Hermite-Hadamard like integral inequalities via these new classes of convex functions. Some special cases are discussed, which can be obtained from our main results. The ideas and techniques of this paper may stimulate further research. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce a new class of M T -preinvex functions, which are called (ξ , m , M T) -preinvex functions, then we give some new estimates of Hermite–Hadamard, Ostrowski and Simpson type inequalities using Riemann–Liouville fractional integral. The obtained results in this paper generalize the well-known results in recent years. [ABSTRACT FROM AUTHOR]