MDPI - Publisher of Open Access Journals

18 pages, 804 KB

Open AccessArticle

New Fractional Hermite–Hadamard-Type Inequalities for Caputo Derivative and MET-(p,s)-Convex Functions with Applications

by Muhammad Sajid Zahoor, Amjad Hussain and Yuanheng Wang

Fractal Fract. 2026, 10(1), 62; https://doi.org/10.3390/fractalfract10010062 - 15 Jan 2026

Abstract

This article investigates fractional Hermite–Hadamard integral inequalities through the framework of Caputo fractional derivatives and MET-

(p, s)

-convex functions. In particular, we introduce new modifications to two classical fractional extensions of Hermite–Hadamard-type inequalities, formulated for both MET- [...] Read more.

This article investigates fractional Hermite–Hadamard integral inequalities through the framework of Caputo fractional derivatives and MET-

(p, s)

-convex functions. In particular, we introduce new modifications to two classical fractional extensions of Hermite–Hadamard-type inequalities, formulated for both MET-

(p, s)

-convex functions and logarithmic

(p, s)

-convex functions. Moreover, we obtain enhancements of inequalities like the Hermite–Hadamard, midpoint, and Fejér types for two extended convex functions by employing the Caputo fractional derivative. The research presents a numerical example with graphical representations to confirm the correctness of the obtained results. Full article

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

24 pages, 502 KB

Open AccessArticle

Deriving Hermite–Hadamard-Type Inequalities via Stochastic k-Caputo Fractional Derivatives

by Ymnah Alruwaily, Raouf Fakhfakh, Ghadah Alomani, Rabab Alzahrani and Abdellatif Ben Makhlouf

Fractal Fract. 2025, 9(12), 757; https://doi.org/10.3390/fractalfract9120757 - 22 Nov 2025

Viewed by 533

Abstract

By leveraging the concept of k-Caputo fractional derivatives for stochastic processes, in this paper, we derive a generalized Hermite–Hadamard inequality tailored to n-times differentiable convex stochastic processes, providing a powerful tool for analyzing systems governed by fractional dynamics in probabilistic settings. [...] Read more.

By leveraging the concept of k-Caputo fractional derivatives for stochastic processes, in this paper, we derive a generalized Hermite–Hadamard inequality tailored to n-times differentiable convex stochastic processes, providing a powerful tool for analyzing systems governed by fractional dynamics in probabilistic settings. Additionally, we establish two new integral identities that serve as the foundation for developing midpoint- and trapezium-type inequalities for

(n + 1)

-times differentiable convex stochastic processes. These results not only enrich the theoretical underpinnings of fractional calculus, but also offer practical implications for modeling and understanding complex systems with memory and randomness. The proposed framework opens new avenues for future research in stochastic analysis and fractional calculus, with potential applications in fields such as financial mathematics, engineering, and physics. Full article

(This article belongs to the Section General Mathematics, Analysis)

► Show Figures

Figure 1

23 pages, 398 KB

Open AccessArticle

On Fractional Hermite–Hadamard-Type Inequalities for Harmonically s-Convex Stochastic Processes

by Rabab Alzahrani, Raouf Fakhfakh, Ghadah Alomani and Badreddine Meftah

Fractal Fract. 2025, 9(11), 750; https://doi.org/10.3390/fractalfract9110750 - 20 Nov 2025

Viewed by 520

Abstract

In this paper, we investigate Hermite–Hadamard-type inequalities for harmonically s-convex stochastic processes via Riemann–Liouville fractional integrals. We begin by introducing the notion of harmonically s-convex stochastic processes. Subsequently, we establish a variety of Riemann–Liouville fractional Hermite–Hadamard-type inequalities for harmonic s-convex [...] Read more.

In this paper, we investigate Hermite–Hadamard-type inequalities for harmonically s-convex stochastic processes via Riemann–Liouville fractional integrals. We begin by introducing the notion of harmonically s-convex stochastic processes. Subsequently, we establish a variety of Riemann–Liouville fractional Hermite–Hadamard-type inequalities for harmonic s-convex stochastic. We first provide the Hermite–Hadamard inequality, then by introducing a novel identity involving mean-square stochastic Riemann–Liouville fractional integral operators, we derive several midpoint-type inequalities for harmonically s-convex stochastic processes. Illustrative example with graphical depiction and a practical application are provided. Full article

(This article belongs to the Section General Mathematics, Analysis)

► Show Figures

Figure 1

21 pages, 1014 KB

Open AccessArticle

Some New Boole-Type Inequalities via Modified Convex Functions with Their Applications and Computational Analysis

by Talha Anwar, Abdul Mateen, Hela Elmannai, Muhammad Aamir Ali and Loredana Ciurdariu

Mathematics 2025, 13(21), 3517; https://doi.org/10.3390/math13213517 - 3 Nov 2025

Viewed by 390

Abstract

In numerical analysis, the Boole’s formula serves as a pivotal tool for approximating definite integrals. The approximation of the definite integrals has a big role in numerical methods for differential equations; in particular, in the finite volume method, we need to use the [...] Read more.

In numerical analysis, the Boole’s formula serves as a pivotal tool for approximating definite integrals. The approximation of the definite integrals has a big role in numerical methods for differential equations; in particular, in the finite volume method, we need to use the best approximation of the integrals to obtain better results. This paper presents a rigorous proof of integral inequalities for first-time differentiable s-convex functions in the second sense. This paper has two main goals. The first is that the use of s-convex function extends the results for convex functions which cover a large class of functions and the second is the best approximation. To prove the main inequalities, we drive integral identity for differentiable functions. Then, with the help of this identity, we prove the error bounds of Boole’s formula for differentiable s-convex functions in the second sense. Some new midpoint-type inequalities for generalized convex functions are also given which can help us in finding better error bounds for midpoint integration formulas compared to the existing ones. Moreover, we provide some applications to quadrature formulas and special means for the real numbers of these newly established inequalities. Furthermore, we present numerical examples and computational analysis that show that these newly established inequalities are numerically valid. Full article

► Show Figures

Figure 1

24 pages, 407 KB

Open AccessArticle

Multiplicative Fractional Hermite–Hadamard-Type Inequalities in G-Calculus

by Abdelghani Lakhdari and Wedad Saleh

Mathematics 2025, 13(21), 3426; https://doi.org/10.3390/math13213426 - 27 Oct 2025

Cited by 2 | Viewed by 433

Abstract

This paper extends Hermite–Hadamard-type inequalities to the fractional multiplicative framework of G-calculus. Using multiplicative Riemann–Liouville fractional integrals, we introduce a notion of multiplicative convexity and establish fractional Hermite–Hadamard, midpoint, and trapezoidal inequalities for

G G

-convex functions. Examples and graphical illustrations are [...] Read more.

This paper extends Hermite–Hadamard-type inequalities to the fractional multiplicative framework of G-calculus. Using multiplicative Riemann–Liouville fractional integrals, we introduce a notion of multiplicative convexity and establish fractional Hermite–Hadamard, midpoint, and trapezoidal inequalities for

G G

-convex functions. Examples and graphical illustrations are provided to demonstrate the applicability of our results and further highlight the role of fractional multiplicative analysis in broadening traditional integral inequalities. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

► Show Figures

Figure 1

28 pages, 869 KB

Open AccessArticle

Local Fractional Perspective on Weddle’s Inequality in Fractal Space

by Yuanheng Wang, Usama Asif, Muhammad Uzair Awan, Muhammad Zakria Javed, Awais Gul Khan, Mona Bin-Asfour and Kholoud Saad Albalawi

Fractal Fract. 2025, 9(10), 662; https://doi.org/10.3390/fractalfract9100662 - 14 Oct 2025

Viewed by 547

Abstract

The Yang local fractional setting provides the generalized framework to explore the non-differentiable mappings considering the local properties. Due to the dominance of these concepts, mathematicians have investigated multiple problems, including mathematical modelling, optimization, and inequalities. Incorporating these useful concepts, this study aims [...] Read more.

The Yang local fractional setting provides the generalized framework to explore the non-differentiable mappings considering the local properties. Due to the dominance of these concepts, mathematicians have investigated multiple problems, including mathematical modelling, optimization, and inequalities. Incorporating these useful concepts, this study aims to derive Weddle-type integral inequalities within the context of fractal space. To achieve the intended results, we establish a new local fractional identity. By using this identity, the convexity property, the bounded property of mappings, the L-Lipschitzian property of mappings, and other famous inequalities, we develop numerous upper bounds. Additionally, we provide 2D and 3D graphical representations and numerous applications, which show the significance of our main findings. To the best of our knowledge, this is the first study concerning error inequalities of Weddle’s quadrature formulation within the fractal space. Full article

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

► Show Figures

Figure 1

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 677

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)

► Show Figures

Figure 1

20 pages, 455 KB

Open AccessArticle

New Estimates of the q-Hermite–Hadamard Inequalities via Strong Convexity

by Chanokgan Sahatsathatsana and Pongsakorn Yotkaew

Axioms 2025, 14(8), 576; https://doi.org/10.3390/axioms14080576 - 25 Jul 2025

Cited by 2 | Viewed by 521

Abstract

A refined version of the q-Hermite–Hadamard inequalities for strongly convex functions is introduced in this paper, utilizing both left and right q-integrals. Tighter bounds and more accurate estimates are derived by incorporating strong convexity. New q-trapezoidal and q-midpoint estimates [...] Read more.

A refined version of the q-Hermite–Hadamard inequalities for strongly convex functions is introduced in this paper, utilizing both left and right q-integrals. Tighter bounds and more accurate estimates are derived by incorporating strong convexity. New q-trapezoidal and q-midpoint estimates are also presented to enhance the precision of the results. The improvements in the results compared to previous work are demonstrated through numerical examples in terms of precision and tighter bounds, and the advantages of using strongly convex functions are showcased. Full article

(This article belongs to the Section Mathematical Analysis)

► Show Figures

Figure 1

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 6 | Viewed by 942

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)

► Show Figures

Figure 1

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 1135

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)

► Show Figures

Figure 1

31 pages, 1687 KB

Open AccessArticle

Some Classical Inequalities Associated with Generic Identity and Applications

by Muhammad Zakria Javed, Muhammad Uzair Awan, Bandar Bin-Mohsin, Hüseyin Budak and Silvestru Sever Dragomir

Axioms 2024, 13(8), 533; https://doi.org/10.3390/axioms13080533 - 6 Aug 2024

Cited by 4 | Viewed by 1387

Abstract

In this paper, we derive a new generic equality for the first-order differentiable functions. Through the utilization of the general identity and convex functions, we produce a family of upper bounds for numerous integral inequalities like Ostrowski’s inequality, trapezoidal inequality, midpoint inequality, Simpson’s [...] Read more.

In this paper, we derive a new generic equality for the first-order differentiable functions. Through the utilization of the general identity and convex functions, we produce a family of upper bounds for numerous integral inequalities like Ostrowski’s inequality, trapezoidal inequality, midpoint inequality, Simpson’s inequality, Newton-type inequalities, and several two-point open trapezoidal inequalities. Also, we provide the numerical and visual explanation of our principal findings. Later, we provide some novel applications to the theory of means, special functions, error bounds of composite quadrature schemes, and parametric iterative schemes to find the roots of linear functions. Also, we attain several already known and new bounds for different values of

γ

and parameter

ξ

. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities)

► Show Figures

Figure 1

24 pages, 376 KB

Open AccessArticle

Further Hermite–Hadamard-Type Inequalities for Fractional Integrals with Exponential Kernels

by Hong Li, Badreddine Meftah, Wedad Saleh, Hongyan Xu, Adem Kiliçman and Abdelghani Lakhdari

Fractal Fract. 2024, 8(6), 345; https://doi.org/10.3390/fractalfract8060345 - 7 Jun 2024

Cited by 14 | Viewed by 1948

Abstract

This paper introduces new versions of Hermite–Hadamard, midpoint- and trapezoid-type inequalities involving fractional integral operators with exponential kernels. We explore these inequalities for differentiable convex functions and demonstrate their connections with classical integrals. This paper validates the derived inequalities through a numerical example [...] Read more.

This paper introduces new versions of Hermite–Hadamard, midpoint- and trapezoid-type inequalities involving fractional integral operators with exponential kernels. We explore these inequalities for differentiable convex functions and demonstrate their connections with classical integrals. This paper validates the derived inequalities through a numerical example with graphical representations and provides some practical applications, highlighting their relevance to special means. This study presents novel results, offering new insights into classical integrals as the fractional order

β

approaches 1, in addition to the fractional integrals we examined. Full article

(This article belongs to the Special Issue New Trends on Generalized Fractional Calculus)

► Show Figures

Figure 1

24 pages, 2145 KB

Open AccessArticle

Analysis and Applications of Some New Fractional Integral Inequalities

by Sofia Ramzan, Muhammad Uzair Awan, Silvestru Sever Dragomir, Bandar Bin-Mohsin and Muhammad Aslam Noor

Fractal Fract. 2023, 7(11), 797; https://doi.org/10.3390/fractalfract7110797 - 31 Oct 2023

Cited by 4 | Viewed by 2437

Abstract

This paper presents a novel parameterized fractional integral identity. By using this auxiliary result and the

s

-convexity property of the mapping, a series of fractional variants of certain classical inequalities, including Simpson’s, midpoint, and trapezoidal-type inequalities, have been derived. Additionally, some applications [...] Read more.

This paper presents a novel parameterized fractional integral identity. By using this auxiliary result and the

s

-convexity property of the mapping, a series of fractional variants of certain classical inequalities, including Simpson’s, midpoint, and trapezoidal-type inequalities, have been derived. Additionally, some applications of our main outcomes to special means of real numbers have been explored. Moreover, we have derived a new generic numerical scheme for solving non-linear equations, demonstrating an application of our main results in numerical analysis. Full article

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

► Show Figures

Figure 1

13 pages, 307 KB

Open AccessArticle

New Fractional Integral Inequalities via k-Atangana–Baleanu Fractional Integral Operators

by Seth Kermausuor and Eze R. Nwaeze

Fractal Fract. 2023, 7(10), 740; https://doi.org/10.3390/fractalfract7100740 - 8 Oct 2023

Cited by 8 | Viewed by 1889

Abstract

We propose the definitions of some fractional integral operators called k-Atangana–Baleanu fractional integral operators. These newly proposed operators are generalizations of the well-known Atangana–Baleanu fractional integral operators. As an application, we establish a generalization of the Hermite–Hadamard inequality. Additionally, we establish some [...] Read more.

We propose the definitions of some fractional integral operators called k-Atangana–Baleanu fractional integral operators. These newly proposed operators are generalizations of the well-known Atangana–Baleanu fractional integral operators. As an application, we establish a generalization of the Hermite–Hadamard inequality. Additionally, we establish some new identities involving these new integral operators and obtained new fractional integral inequalities of the midpoint and trapezoidal type for functions whose derivatives are bounded or convex. Full article

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

19 pages, 551 KB

Open AccessArticle

Improvement in Some Inequalities via Jensen–Mercer Inequality and Fractional Extended Riemann–Liouville Integrals

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

Axioms 2023, 12(9), 886; https://doi.org/10.3390/axioms12090886 - 17 Sep 2023

Cited by 4 | Viewed by 1539

Abstract

The primary intent of this study is to establish some important inequalities of the Hermite–Hadamard, trapezoid, and midpoint types under fractional extended Riemann–Liouville integrals (FERLIs). The proofs are constructed using the renowned Jensen–Mercer, power-mean, and Holder inequalities. Various equalities for the FERLIs and [...] Read more.

The primary intent of this study is to establish some important inequalities of the Hermite–Hadamard, trapezoid, and midpoint types under fractional extended Riemann–Liouville integrals (FERLIs). The proofs are constructed using the renowned Jensen–Mercer, power-mean, and Holder inequalities. Various equalities for the FERLIs and convex functions are construed to be the mainstay for finding new results. Some connections between our main findings and previous research on Riemann–Liouville fractional integrals and FERLIs are also discussed. Moreover, a number of examples are featured, with graphical representations to illustrate and validate the accuracy of the new findings. Full article

(This article belongs to the Special Issue Fractional Calculus - Theory and Applications II)

► Show Figures

Figure 1

Search Results (58)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (58)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI