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

Authors

Fuxiang Liu, Jielan Li

Type: Article

Publication Date: 2025-05-04

Citations: 0

DOI: https://doi.org/10.3390/sym17050702

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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 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.

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Symmetry
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2022-01-13

Soubhagya Kumar Sahoo , Muhammad Tariq , Hijaz Ahmad +4 more

The comprehension of inequalities in convexity is very important for fractional calculus and its effectiveness in many applied sciences. In this article, we handle a novel investigation that depends on … The comprehension of inequalities in convexity is very important for fractional calculus and its effectiveness in many applied sciences. In this article, we handle a novel investigation that depends on the Hermite–Hadamard-type inequalities concerning a monotonic increasing function. The proposed methodology deals with a new class of convexity and related integral and fractional inequalities. There exists a solid connection between fractional operators and convexity because of its fascinating nature in the numerical sciences. Some special cases have also been discussed, and several already-known inequalities have been recaptured to behave well. Some applications related to special means, q-digamma, modified Bessel functions, and matrices are discussed as well. The aftereffects of the plan show that the methodology can be applied directly and is computationally easy to understand and exact. We believe our findings generalise some well-known results in the literature on s-convexity.

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On some Simpson's and Newton's type of inequalities in multiplicative calculus with applications

2022-11-29

Saowaluck Chasreechai , Muhammad Aamir Ali , Surapol Naowarat +2 more

<abstract>In this paper, we establish an integral equality involving a multiplicative differentiable function for the multiplicative integral. Then, we use the newly established equality to prove some new Simpson's and … <abstract>In this paper, we establish an integral equality involving a multiplicative differentiable function for the multiplicative integral. Then, we use the newly established equality to prove some new Simpson's and Newton's inequalities for multiplicative differentiable functions. Finally, we give some mathematical examples to show the validation of newly established inequalities.</abstract>

On Riemann-Type Weighted Fractional Operators and Solutions to Cauchy Problems

2023-01-01

Muhammad Samraiz , Muhammad Umer , Thabet Abdeljawad +3 more

In this paper, we establish the new forms of Riemann-type fractional integral and derivative operators. The novel fractional integral operator is proved to be bounded in Lebesgue space and some … In this paper, we establish the new forms of Riemann-type fractional integral and derivative operators. The novel fractional integral operator is proved to be bounded in Lebesgue space and some classical fractional integral and differential operators are obtained as special cases. The properties of new operators like semi-group, inverse and certain others are discussed and its weighted Laplace transform is evaluated. Fractional integro-differential free-electron laser (FEL) and kinetic equations are established. The solutions to these new equations are obtained by using the modified weighted Laplace transform. The Cauchy problem and a growth model are designed as applications along with graphical representation. Finally, the conclusion section indicates future directions to the readers.

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Stability analysis of a non-singular fractional-order covid-19 model with nonlinear incidence and treatment rate

2023-02-23

Hardik Joshi , Mehmet Yavuz , Stuart Townley +1 more

Abstract In this paper, a non-singular SIR model with the Mittag-Leffler law is proposed. The nonlinear Beddington-DeAngelis infection rate and Holling type II treatment rate are used. The qualitative properties … Abstract In this paper, a non-singular SIR model with the Mittag-Leffler law is proposed. The nonlinear Beddington-DeAngelis infection rate and Holling type II treatment rate are used. The qualitative properties of the SIR model are discussed in detail. The local and global stability of the model are analyzed. Moreover, some conditions are developed to guarantee local and global asymptotic stability. Finally, numerical simulations are provided to support the theoretical results and used to analyze the impact of face masks, social distancing, quarantine, lockdown, immigration, treatment rate of the disease, and limitation in treatment resources on COVID-19. The graphical results show that face masks, social distancing, quarantine, lockdown, immigration, and effective treatment rates significantly reduce the infected population over time. In contrast, limitation in the availability of treatment raises the infected population.

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Some New Hermite–Hadamard Type Inequalities Pertaining to Generalized Multiplicative Fractional Integrals

2023-04-05

Artion Kashuri , Soubhagya Kumar Sahoo , Munirah Aljuaid +2 more

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Some new fractional Hermite-Hadamard type inequalities for functions with co-ordinated extended <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mrow><mml:mrow><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mrow></mml:math>-prequasiinvex mixed partial derivatives

2023-04-12

Wedad Saleh , Abdelghani Lakhdari , Adem Kılıçman +2 more

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

2023-01-01

Fiza Zafar , Sikander Mehmood , Asim Asiri

Abstract In this article, we have established some new bounds of Fejér-type Hermite-Hadamard inequality for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> </m:math> k -fractional integrals involving <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>r</m:mi> </m:math> r -times differentiable … Abstract In this article, we have established some new bounds of Fejér-type Hermite-Hadamard inequality for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> </m:math> k -fractional integrals involving <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>r</m:mi> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> </m:math> k -fractional integrals for generalized convex functions available in the literature.

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

2023-09-01

Junxi Chen , Chunyan Luo

Hermite–Hadamard type inequalities for multiplicative Riemann–Liouville fractional integrals

2023-09-22

Tingsong Du , Yu Peng

Hermite–Hadamard type inequalities for multiplicatively harmonic convex functions

2023-09-22

Serap Özcan , Saad Ihsan Butt

Abstract In this work, the notion of a multiplicative harmonic convex function is examined, and Hermite–Hadamard inequalities for this class of functions are established. Many inequalities of Hermite–Hadamard type are … Abstract In this work, the notion of a multiplicative harmonic convex function is examined, and Hermite–Hadamard inequalities for this class of functions are established. Many inequalities of Hermite–Hadamard type are also taken into account for the product and quotient of multiplicative harmonic convex functions. In addition, new multiplicative integral-based inequalities are found for the quotient and product of multiplicative harmonic convex and harmonic convex functions. In addition, we provide certain upper limits for such classes of functions. The obtained results have been verified by providing examples with included graphs. The findings of this study may encourage more research in several scientific areas.

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Hermite-Hadamard type inequalities for multiplicatively p-convex functions

2023-09-26

Serap Özcan

Abstract In this paper, we introduced the concept of multiplicatively p -convex functions and established Hermite-Hadamard type integral inequalities in the setting of multiplicative calculus for this newly created class … Abstract In this paper, we introduced the concept of multiplicatively p -convex functions and established Hermite-Hadamard type integral inequalities in the setting of multiplicative calculus for this newly created class of functions. We also gave integral inequalities of Hermite-Hadamard type for product and quotient of multiplicatively p -convex functions. Furthermore, we obtained novel multiplicative integral-based inequalities for the product and quotient of convex and multiplicatively p -convex functions. Additionally, we derived certain upper limits for this new class of functions. The findings we proved are generalizations of the results in the literature. The results obtained in this study may inspire further research in various scientific areas.

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On Some New AB-Fractional Inclusion Relations

2023-09-30

Bandar Bin‐Mohsin , Muhammad Zakria Javed , Muhammad Uzair Awan +1 more

The theory of integral inequality has gained considerable attention due to its influential impact on several fields of mathematics and applied sciences. Over the years, numerous refinements, generalizations, and extensions … The theory of integral inequality has gained considerable attention due to its influential impact on several fields of mathematics and applied sciences. Over the years, numerous refinements, generalizations, and extensions of convexity have been explored to achieve more precise variants of already established results. The principal idea of this article is to establish some interval-valued integral inequalities of the Hermite–Hadamard type in the fractional domain. First, we propose the idea of generalized interval-valued convexity with respect to the continuous monotonic functions ⋎, bifunction ζ, and based on the containment ordering relation, which is termed as (⋎,h) pre-invex functions. This class is innovative due to its generic characteristics. We generate numerous known and new classes of convexity by considering various values for ⋎ and h. Moreover, we use the notion of (⋎,h)-pre-invexity and Atangana–Baleanu (AB) fractional operators to develop some fresh fractional variants of the Hermite–Hadamard (HH), Pachpatte, and Hermite–Hadamard–Fejer (HHF) types of inequalities. The outcomes obtained here are the most unified forms of existing results. We provide several specific cases, as well as a numerical and graphical study, to show the significance of the major results.

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Optimizing signal smoothing using HERS algorithm and time fractional diffusion equation

2023-11-01

Amutha Praba Jayaraj , Kuppuswamy Nallappa Gounder , Jeetendra Rajagopal

On multiplicative Hermite–Hadamard- and Newton-type inequalities for multiplicatively (P,m)-convex functions

2024-01-13

Lulu Zhang , Yu Peng , Tingsong Du

Symmetrical Hermite–Hadamard type inequalities stemming from multiplicative fractional integrals

2024-05-08

Yu Peng , Serap Özcan , Tingsong Du

Some new midpoint and trapezoidal type inequalities in multiplicative calculus with applications

2023-01-01

Jianqiang Xie , Ali Muhammad , Sitthiwirattham Thanin

In this paper, we use multiplicative twice differentiable functions and establish two new multiplicative integral identities. Then, we use convexity for multiplicative twice differentiable functions and establish some new midpoint … In this paper, we use multiplicative twice differentiable functions and establish two new multiplicative integral identities. Then, we use convexity for multiplicative twice differentiable functions and establish some new midpoint and trapezoidal type inequalities in the framework of multiplicative calculus. Finally, we give some applications to special means of real numbers to make these inequalities more interesting for the readers.

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Dual Simpson type inequalities for multiplicatively convex functions

2023-01-01

Badreddine Meftah , Abdelghani Lakhdari

In this paper we propose a new identity for multiplicative differentiable functions, based on this identity we establish a dual Simpson type inequality for multiplicatively convex functions. Some applications of … In this paper we propose a new identity for multiplicative differentiable functions, based on this identity we establish a dual Simpson type inequality for multiplicatively convex functions. Some applications of the obtained results are also given.

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Hermite-Hadamard type inequalities for exponential type multiplicatively convex functions

2023-01-01

Serap Özcan

In this paper, we defined and studied the concept of exponential type multiplicatively convex functions and some of their algebraic properties. We derived Hermite-Hadamard inequalities for this class of functions. … In this paper, we defined and studied the concept of exponential type multiplicatively convex functions and some of their algebraic properties. We derived Hermite-Hadamard inequalities for this class of functions. We also established new Hermite-Hadamard type inequalities for the product and quotient of exponential type multiplicatively convex functions. In addition, we obtained new multiplicative integral based inequalities for the quotient and product of exponential type multiplicatively convex functions and convex functions. The results in this study could potentially inspire further research in various scientific fields.

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On Extended Class of Totally Ordered Interval-Valued Convex Stochastic Processes and Applications

2024-09-30

Muhammad Zakria Javed , Muhammad Uzair Awan , Loredana Ciurdariu +2 more

The intent of the current study is to explore convex stochastic processes within a broader context. We introduce the concept of unified stochastic processes to analyze both convex and non-convex … The intent of the current study is to explore convex stochastic processes within a broader context. We introduce the concept of unified stochastic processes to analyze both convex and non-convex stochastic processes simultaneously. We employ weighted quasi-mean, non-negative mapping γ, and center-radius ordering relations to establish a class of extended cr-interval-valued convex stochastic processes. This class yields a combination of innovative convex and non-convex stochastic processes. We characterize our class by illustrating its relationships with other classes as well as certain key attributes and sufficient conditions for this class of processes. Additionally, leveraging Riemann–Liouville stochastic fractional operators and our proposed class, we prove parametric fractional variants of Jensen’s inequality, Hermite–Hadamard’s inequality, Fejer’s inequality, and product Hermite–Hadamard’s like inequality. We establish an interesting relation between means by means of Hermite–Hadamard’s inequality. We utilize the numerical and graphical approaches to showcase the significance and effectiveness of primary findings. Also, the proposed results are powerful tools to evaluate the bounds for stochastic Riemann–Liouville fractional operators in different scenarios for a larger space of processes.