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33 pages, 1577 KB

Open AccessArticle

Refined Hermite–Hadamard Type Inequalities via Multiplicative Non-Singular Fractional Integral Operators and Applications in Superquadratic Structures

by Ghulam Jallani, Saad Ihsan Butt, Dawood Khan and Youngsoo Seol

Fractal Fract. 2025, 9(9), 617; https://doi.org/10.3390/fractalfract9090617 - 22 Sep 2025

Viewed by 148

Abstract

The aim of this manuscript is to introduce the fractional integral inequalities of H-H types via multiplicative (Antagana-Baleanu) A-B fractional operators. We also provide the fractional version of the H-H type of the product and quotient of multiplicative superquadratic and multiplicative subquadratic functions [...] Read more.

The aim of this manuscript is to introduce the fractional integral inequalities of H-H types via multiplicative (Antagana-Baleanu) A-B fractional operators. We also provide the fractional version of the H-H type of the product and quotient of multiplicative superquadratic and multiplicative subquadratic functions via the same operators. Superquadratic functions, have stronger convexity-like behavior. They provide sharper bounds and more refined inequalities, which are valuable in optimization, information theory, and related fields. The use of multiplicative fractional operators establishes a nonlinear fractional structure, enhancing the analytical tools available for studying dynamic and nonlinear systems. The authenticity of the obtained results are verified by graphical and numerical illustrations by taking into account some examples. Additionally, the study explores applications involving special means, special functions and moments of random variables resulting in new fractional recurrence relations within the multiplicative calculus framework. These contributions not only generalize existing inequalities but also pave the way for future research in both theoretical mathematics and real-world modeling scenarios. Full article

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26 pages, 717 KB

Open AccessArticle

Evolutionary Approach to Inequalities of Hermite–Hadamard–Mercer Type for Generalized Wright’s Functions Associated with Computational Evaluation and Their Applications

by Talib Hussain, Loredana Ciurdariu and Eugenia Grecu

Fractal Fract. 2025, 9(9), 593; https://doi.org/10.3390/fractalfract9090593 - 10 Sep 2025

Viewed by 320

Abstract

The theory of integral inequalities has a wide range of applications in physics and numerical computation, and plays a fundamental role in mathematical analysis. The present study delves into the attractive domain of Hermite–Hadamard–Mercer (H–H–M)-type inequalities having a special emphasis on Wright’s general [...] Read more.

The theory of integral inequalities has a wide range of applications in physics and numerical computation, and plays a fundamental role in mathematical analysis. The present study delves into the attractive domain of Hermite–Hadamard–Mercer (H–H–M)-type inequalities having a special emphasis on Wright’s general functions, referred to as Raina’s functions in the scientific literature. The main goal of our progressive study is to use Raina’s Fractional Integrals to derive two useful lemmas for second-differentiable functions. Using the derived lemmas, we proved a large number of fractional integral inequalities related to trapezoidal and midpoint-type inequalities where those that are twice differentiable in absolute values are convex. Some of these results also generalize findings from previous research. Next, we provide applications to error estimates for trapezoidal and midpoint quadrature formulas and to analytical evaluations involving modified Bessel functions of the first kind and q-digamma functions, and we show the validity of the proposed inequalities in numerical integration and analysis of special functions. Finally, the results are well-supported by numerous examples, including graphical representations and numerical tables, which collectively highlight their accuracy and computational significance. Full article

(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 3rd Edition)

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22 pages, 805 KB

Open AccessArticle

A Symmetric Quantum Perspective of Analytical Inequalities and Their Applications

by Muhammad Zakria Javed, Nimra Naeem, Muhammad Uzair Awan, Yuanheng Wang and Omar Mutab Alsalami

Mathematics 2025, 13(18), 2910; https://doi.org/10.3390/math13182910 - 9 Sep 2025

Viewed by 265

Abstract

This study explores some new symmetric quantum inequalities that are based on Breckner’s convexity. By using these concepts, we propose new versions of Hermite–Hadamard (H-H) and Fejer-type inequalities. Additionally, we establish a new integral identity which helped us to derive a set of [...] Read more.

This study explores some new symmetric quantum inequalities that are based on Breckner’s convexity. By using these concepts, we propose new versions of Hermite–Hadamard (H-H) and Fejer-type inequalities. Additionally, we establish a new integral identity which helped us to derive a set of new quantum inequalities. Using the symmetric quantum identity, Breckner’s convexity, and several other classical inequalities, we develop blended bounds for a general quadrature scheme. To ensure the significance of this study, a few captivating applications are discussed. Full article

(This article belongs to the Special Issue Mathematical Inequalities and Fractional Calculus)

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23 pages, 419 KB

Open AccessArticle

Hermite–Hadamard-Type Inequalities for h-Godunova–Levin Convex Fuzzy Interval-Valued Functions via Riemann–Liouville Fractional q-Integrals

by Muhammad Waseem Akram, Sajid Iqbal, Asfand Fahad and Yuanheng Wang

Fractal Fract. 2025, 9(9), 578; https://doi.org/10.3390/fractalfract9090578 - 31 Aug 2025

Viewed by 351

Abstract

In this study, we develop new Hermite–Hadamard and Hermite–Hadamard–Fejér type inequalities for fuzzy interval-valued functions (FIVFs) that exhibit h-Godunova–Levin convexity, using the framework of the Riemann–Liouville fractional (RLF) q-integral. We introduce novel fuzzy extensions of classical inequalities and establish corresponding inclusion [...] Read more.

In this study, we develop new Hermite–Hadamard and Hermite–Hadamard–Fejér type inequalities for fuzzy interval-valued functions (FIVFs) that exhibit h-Godunova–Levin convexity, using the framework of the Riemann–Liouville fractional (RLF) q-integral. We introduce novel fuzzy extensions of classical inequalities and establish corresponding inclusion relations by utilizing the properties of fuzzy RLF q-integrals. Furthermore, we validate the theoretical results through illustrative numerical examples and graphical representations, demonstrating the applicability and effectiveness of the derived inequalities in the context of fuzzy and interval analysis. Full article

(This article belongs to the Special Issue Advances in Fractional Integral Inequalities: Theory and Applications)

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25 pages, 343 KB

Open AccessArticle

Hermite–Hadamard-Mercer Type Inequalities for Interval-Valued Coordinated Convex Functions

by Muhammad Toseef, Iram Javed, Muhammad Aamir Ali and Loredana Ciurdariu

Axioms 2025, 14(9), 661; https://doi.org/10.3390/axioms14090661 - 28 Aug 2025

Viewed by 399

Abstract

Determining the Jensen–Mercer inequality for interval-valued coordinated convex functions has been a challenging task for researchers in the fields of inequalities and interval analysis. We use

⊖_{g}

to establish the Jensen–Mercer inequality for interval-valued coordinated convex functions. In this paper, we make [...] Read more.

Determining the Jensen–Mercer inequality for interval-valued coordinated convex functions has been a challenging task for researchers in the fields of inequalities and interval analysis. We use

⊖_{g}

to establish the Jensen–Mercer inequality for interval-valued coordinated convex functions. In this paper, we make significant strides in establishing new results by introducing a novel approach. We present a Hermite–Hadamard (H.H.) Mercer-type inequality for interval-valued coordinated convex functions and show how it generalizes the traditional H.H. inequality. Specifically, the H.H. inequality for interval-valued coordinated convex functions can be derived as a special case by considering the endpoints of the H.H. Mercer-type inequality. Furthermore, we provide computational results that verify the accuracy of recent findings in the literature. Our results indicate that the proposed new results impose highly effective constraints on integrals of the specified functions and are valid for a broader class of functions. These new findings have significant implications for applications in fields such as economics, engineering, and physics, where they can improve the precision of system modeling and optimization. Full article

(This article belongs to the Section Mathematical Analysis)

37 pages, 776 KB

Open AccessArticle

Fractional Inclusion Analysis of Superquadratic Stochastic Processes via Center-Radius Total Order Relation with Applications in Information Theory

by Mohsen Ayyash, Dawood Khan, Saad Ihsan Butt and Youngsoo Seol

Fractal Fract. 2025, 9(6), 375; https://doi.org/10.3390/fractalfract9060375 - 12 Jun 2025

Cited by 1 | Viewed by 445

Abstract

This study presents, for the first time, a new class of interval-valued superquadratic stochastic processes and examines their core properties through the lens of the center-radius total order relation on intervals. These processes serve as a powerful tool for modeling uncertainty in stochastic [...] Read more.

This study presents, for the first time, a new class of interval-valued superquadratic stochastic processes and examines their core properties through the lens of the center-radius total order relation on intervals. These processes serve as a powerful tool for modeling uncertainty in stochastic systems involving interval-valued data. By utilizing their intrinsic structure, we derive sharpened versions of Jensen-type and Hermite–Hadamard-type inequalities, along with their fractional extensions, within the framework of mean-square stochastic Riemann–Liouville fractional integrals. The theoretical findings are validated through extensive graphical representations and numerical simulations. Moreover, the applicability of the proposed processes is demonstrated in the domain of information theory by constructing novel stochastic divergence measures and Shannon’s entropy grounded in interval calculus. The outcomes of this work lay a solid foundation for further exploration in stochastic analysis, particularly in advancing generalized integral inequalities and formulating new stochastic models under uncertainty. Full article

(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications, 2nd Edition)

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28 pages, 603 KB

Open AccessArticle

New Results on Majorized Discrete Jensen–Mercer Inequality for Raina Fractional Operators

by Çetin Yildiz, Tevfik İşleyen and Luminiţa-Ioana Cotîrlă

Fractal Fract. 2025, 9(6), 343; https://doi.org/10.3390/fractalfract9060343 - 26 May 2025

Viewed by 395

Abstract

As the most important inequality, the Hermite–Hadamard–Mercer inequality has attracted the interest of numerous additional mathematicians. Numerous findings on this inequality have been developed in recent years. So, in this paper, we demonstrate novel Hermite–Hadamard–Mercer inequalities using Raina fractional operators and the majorization [...] Read more.

As the most important inequality, the Hermite–Hadamard–Mercer inequality has attracted the interest of numerous additional mathematicians. Numerous findings on this inequality have been developed in recent years. So, in this paper, we demonstrate novel Hermite–Hadamard–Mercer inequalities using Raina fractional operators and the majorization concept. Furthermore, additional identities are discovered, and two new lemmas of this type are proved. A summary of several known results is also provided, along with a thorough derivation of some exceptional cases. We also note that some of the outcomes in this study are more acceptable than others under certain exceptional instances, such as setting

n = 2

,

w = 0

,

σ (0) = 1

, and

λ = 1

or

λ = α

. Lastly, the method described in this publication is thought to stimulate further research in this area. Full article

(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 3rd Edition)

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34 pages, 437 KB

Open AccessArticle

On Katugampola Fractional Multiplicative Hermite-Hadamard-Type Inequalities

by Wedad Saleh, Badreddine Meftah, Muhammad Uzair Awan and Abdelghani Lakhdari

Mathematics 2025, 13(10), 1575; https://doi.org/10.3390/math13101575 - 10 May 2025

Cited by 1 | Viewed by 566

Abstract

This paper presents a novel framework for Katugampola fractional multiplicative integrals, advancing recent breakthroughs in fractional calculus through a synergistic integration of multiplicative analysis. Motivated by the growing interest in fractional calculus and its applications, we address the gap in generalized inequalities for [...] Read more.

This paper presents a novel framework for Katugampola fractional multiplicative integrals, advancing recent breakthroughs in fractional calculus through a synergistic integration of multiplicative analysis. Motivated by the growing interest in fractional calculus and its applications, we address the gap in generalized inequalities for multiplicative s-convex functions by deriving a Hermite–Hadamard-type inequality tailored to Katugampola fractional multiplicative integrals. A cornerstone of our work involves the derivation of two groundbreaking identities, which serve as the foundation for midpoint- and trapezoid-type inequalities designed explicitly for mappings whose multiplicative derivatives are multiplicative s-convex. These results extend classical integral inequalities to the multiplicative fractional calculus setting, offering enhanced precision in approximating nonlinear phenomena. Full article

(This article belongs to the Special Issue Mathematical Inequalities and Fractional Calculus)

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18 pages, 288 KB

Open AccessFeature PaperArticle

Majorization-Type Integral Inequalities Related to a Result of Bennett with Applications

by László Horváth

Mathematics 2025, 13(10), 1563; https://doi.org/10.3390/math13101563 - 9 May 2025

Viewed by 354

Abstract

In this paper, starting from abstract versions of a result of Bennett given by Niculescu, we derive new majorization-type integral inequalities for convex functions using finite signed measures. The proof of the main result is based on a generalization of a recently discovered [...] Read more.

In this paper, starting from abstract versions of a result of Bennett given by Niculescu, we derive new majorization-type integral inequalities for convex functions using finite signed measures. The proof of the main result is based on a generalization of a recently discovered majorization-type integral inequality. As applications of the results, we give simple proofs of the integral Jensen and Lah–Ribarič inequalities for finite signed measures, generalize and extend known results, and obtain an interesting new refinement of the Hermite–Hadamard–Fejér inequality. Full article

32 pages, 6743 KB

Open AccessArticle

Analytical Properties and Hermite–Hadamard Type Inequalities Derived from Multiplicative Generalized Proportional σ-Riemann–Liouville Fractional Integrals

by Fuxiang Liu and Jielan Li

Symmetry 2025, 17(5), 702; https://doi.org/10.3390/sym17050702 - 4 May 2025

Cited by 1 | Viewed by 548

Abstract

This paper investigates the analytical properties of multiplicative generalized proportional

σ

-Riemann–Liouville fractional integrals and the corresponding Hermite–Hadamard-type inequalities. Central to our study are two key notions: multiplicative

σ

-convex functions and multiplicative generalized proportional

σ

-Riemann–Liouville fractional integrals, both of which serve [...] Read more.

This paper investigates the analytical properties of multiplicative generalized proportional

σ

-Riemann–Liouville fractional integrals and the corresponding Hermite–Hadamard-type inequalities. Central to our study are two key notions: multiplicative

σ

-convex functions and multiplicative generalized proportional

σ

-Riemann–Liouville fractional integrals, both of which serve as the foundational framework for our analysis. We first introduce and examine several fundamental properties of the newly defined fractional integral operator, including continuity, commutativity, semigroup behavior, and boundedness. Building on these results, we derive a novel identity involving this operator, which forms the basis for establishing new Hermite–Hadamard-type inequalities within the multiplicative setting. To validate the theoretical results, we provide multiple illustrative examples and perform graphical visualizations. These examples not only demonstrate the correctness of the derived inequalities but also highlight the practical relevance and potential applications of the proposed framework. Full article

(This article belongs to the Section Mathematics)

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30 pages, 595 KB

Open AccessArticle

New Perspectives of Hermite–Hadamard–Mercer-Type Inequalities Associated with ψ_k-Raina’s Fractional Integrals for Differentiable Convex Functions

by Talib Hussain, Loredana Ciurdariu and Eugenia Grecu

Fractal Fract. 2025, 9(4), 203; https://doi.org/10.3390/fractalfract9040203 - 26 Mar 2025

Cited by 1 | Viewed by 479

Abstract

Starting from

ψ_{k}

-Raina’s fractional integrals (

ψ_{k}

-RFIs), the study obtains a new generalization of the Hermite–Hadamard–Mercer (H-H-M) inequality. Several trapezoid-type inequalities are constructed for functions whose derivatives of orders 1 and 2, in absolute value, are convex and involve [...] Read more.

Starting from

ψ_{k}

-Raina’s fractional integrals (

ψ_{k}

-RFIs), the study obtains a new generalization of the Hermite–Hadamard–Mercer (H-H-M) inequality. Several trapezoid-type inequalities are constructed for functions whose derivatives of orders 1 and 2, in absolute value, are convex and involve

ψ_{k}

-RFIs. The results of the research are refinements of the Hermite–Hadamard (H-H) and H-H-M-type inequalities. For several types of fractional integrals—Riemann–Liouville (R-L), k-Riemann–Liouville (k-R-L),

ψ

-Riemann–Liouville (

ψ

-R-L),

ψ_{k}

-Riemann–Liouville (

ψ_{k}

-R-L), Raina’s, k-Raina’s, and

ψ

-Raina’s fractional integrals (

ψ

-RFIs)—new inequalities of H-H and H-H-M-type are established, respectively. This article presents special cases of the main results and provides numerous examples with graphical illustrations to confirm the validity of the results. This study shows the efficiency of the findings with a couple of applications, taking into account the modified Bessel function and the q-digamma function. Full article

(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications, 2nd Edition)

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19 pages, 370 KB

Open AccessArticle

On Quantum Hermite-Hadamard-Fejer Type Integral Inequalities via Uniformly Convex Functions

by Hasan Barsam, Somayeh Mirzadeh, Yamin Sayyari and Loredana Ciurdariu

Fractal Fract. 2025, 9(2), 108; https://doi.org/10.3390/fractalfract9020108 - 12 Feb 2025

Cited by 2 | Viewed by 930

Abstract

The main goal of this study is to provide new q-Fejer and q-Hermite-Hadamard type integral inequalities for uniformly convex functions and functions whose second quantum derivatives in absolute values are uniformly convex. Two basic inequalities as power mean inequality and Holder’s [...] Read more.

The main goal of this study is to provide new q-Fejer and q-Hermite-Hadamard type integral inequalities for uniformly convex functions and functions whose second quantum derivatives in absolute values are uniformly convex. Two basic inequalities as power mean inequality and Holder’s inequality are used in demonstrations. Some particular functions are chosen to illustrate the investigated results by two examples analyzed and the result obtained have been graphically visualized. Full article

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14 pages, 297 KB

Open AccessArticle

Generalized Fractional Integral Inequalities Derived from Convexity Properties of Twice-Differentiable Functions

by Areej A. Almoneef, Abd-Allah Hyder, Fatih Hezenci and Hüseyin Budak

Fractal Fract. 2025, 9(2), 97; https://doi.org/10.3390/fractalfract9020097 - 4 Feb 2025

Viewed by 893

Abstract

This study presents novel formulations of fractional integral inequalities, formulated using generalized fractional integral operators and the exploration of convexity properties. A key identity is established for twice-differentiable functions with the absolute value of their second derivative being convex. Using this identity, several [...] Read more.

This study presents novel formulations of fractional integral inequalities, formulated using generalized fractional integral operators and the exploration of convexity properties. A key identity is established for twice-differentiable functions with the absolute value of their second derivative being convex. Using this identity, several generalized fractional Hermite–Hadamard-type inequalities are developed. These inequalities extend the classical midpoint and trapezoidal-type inequalities, while offering new perspectives through convexity properties. Also, some special cases align with known results, and an illustrative example, accompanied by a graphical representation, is provided to demonstrate the practical relevance of the results. Moreover, the findings may offer potential applications in numerical integration, optimization, and fractional differential equations, illustrating their relevance to various areas of mathematical analysis. Full article

(This article belongs to the Special Issue New Trends on Generalized Fractional Calculus, 2nd Edition)

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25 pages, 437 KB

Open AccessArticle

Hermite–Hadamard-Type Inequalities for Harmonically Convex Functions via Proportional Caputo-Hybrid Operators with Applications

by Saad Ihsan Butt, Muhammad Umar, Dawood Khan, Youngsoo Seol and Sanja Tipurić-Spužević

Fractal Fract. 2025, 9(2), 77; https://doi.org/10.3390/fractalfract9020077 - 24 Jan 2025

Cited by 1 | Viewed by 1137

Abstract

In this paper, we aim to establish new inequalities of Hermite–Hadamard (H.H) type for harmonically convex functions using proportional Caputo-Hybrid (P.C.H) fractional operators. Parameterized by

α

, these operators offer a unique flexibility: setting

α = 1

recovers the classical inequalities for harmonically [...] Read more.

In this paper, we aim to establish new inequalities of Hermite–Hadamard (H.H) type for harmonically convex functions using proportional Caputo-Hybrid (P.C.H) fractional operators. Parameterized by

α

, these operators offer a unique flexibility: setting

α = 1

recovers the classical inequalities for harmonically convex functions, while setting

α = 0

yields inequalities for differentiable harmonically convex functions. This framework allows us to unify classical and fractional cases within a single operator. To validate the theoretical results, we provide several illustrative examples supported by graphical representations, marking the first use of such visualizations for inequalities derived via P.C.H operators. Additionally, we demonstrate practical applications of the results by deriving new fractional-order recurrence relations for the modified Bessel function of type-1, which are useful in mathematical modeling, engineering, and physics. The findings contribute to the growing body of research in fractional inequalities and harmonic convexity, paving the way for further exploration of generalized convexities and higher-order fractional operators. Full article

(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications, 2nd Edition)

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21 pages, 483 KB

Open AccessArticle

New Inequalities for GA–h Convex Functions via Generalized Fractional Integral Operators with Applications to Entropy and Mean Inequalities

by Asfand Fahad, Zammad Ali, Shigeru Furuichi, Saad Ihsan Butt, Ayesha and Yuanheng Wang

Fractal Fract. 2024, 8(12), 728; https://doi.org/10.3390/fractalfract8120728 - 12 Dec 2024

Cited by 2 | Viewed by 1000

Abstract

We prove the inequalities of the weighted Hermite–Hadamard type the and Hermite–Hadamard–Mercer type for an extremely rich class of geometrically arithmetically-h-convex functions (GA-h-CFs) via generalized Hadamard–Fractional integral operators (HFIOs). The two generalized fractional integral operators (FIOs) are Hadamard proportional [...] Read more.

We prove the inequalities of the weighted Hermite–Hadamard type the and Hermite–Hadamard–Mercer type for an extremely rich class of geometrically arithmetically-h-convex functions (GA-h-CFs) via generalized Hadamard–Fractional integral operators (HFIOs). The two generalized fractional integral operators (FIOs) are Hadamard proportional fractional integral operators (HPFIOs) and Hadamard k-fractional integral operators (HKFIOs). Moreover, we also present the results for subclasses of GA-h-CFs and show that the inequalities proved in this paper unify the results from the recent related literature. Furthermore, we compare the two generalizations in view of the fractional operator parameters that contribute to the generalizations of the results and assess the better approximation via graphical tools. Finally, we present applications of the new inequalities via HPFIOs and HKFIOs by establishing interpolation relations between arithmetic mean and geometric mean and by proving the new upper bounds for the Tsallis relative operator entropy. Full article

(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 3rd Edition)

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