This paper investigates the analytical properties of multiplicative generalized proportional Ļ-RiemannāLiouville fractional integrals and the corresponding HermiteāHadamard-type inequalities. Central to our study are two key notions: multiplicative Ļ-convex functions and ā¦
This paper investigates the analytical properties of multiplicative generalized proportional Ļ-RiemannāLiouville fractional integrals and the corresponding HermiteāHadamard-type inequalities. Central to our study are two key notions: multiplicative Ļ-convex functions and multiplicative generalized proportional Ļ-RiemannāLiouville 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 HermiteāHadamard-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.
This paper investigates the analytical properties of multiplicative generalized proportional Ļ-RiemannāLiouville fractional integrals and the corresponding HermiteāHadamard-type inequalities. Central to our study are two key notions: multiplicative Ļ-convex functions and ā¦
This paper investigates the analytical properties of multiplicative generalized proportional Ļ-RiemannāLiouville fractional integrals and the corresponding HermiteāHadamard-type inequalities. Central to our study are two key notions: multiplicative Ļ-convex functions and multiplicative generalized proportional Ļ-RiemannāLiouville 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 HermiteāHadamard-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.
We give a new definition of fractional derivative and fractional integral. The form of the definition shows that it is the most natural definition, and the most fruitful one. The ā¦
We give a new definition of fractional derivative and fractional integral. The form of the definition shows that it is the most natural definition, and the most fruitful one. The definition for 0ā¤Ī±<1 coincides with the classical definitions on polynomials (up to a constant). Further, if Ī±=1, the definition coincides with the classical definition of first derivative. We give some applications to fractional differential equations.
We develop a parallel theory to the classical theory of convex functions, based on a change of variable formula, by replacing the arithmetic mean by the geometric one.It is shown ā¦
We develop a parallel theory to the classical theory of convex functions, based on a change of variable formula, by replacing the arithmetic mean by the geometric one.It is shown that many interesting functions such as exp, sinh, cosh, sec, csc, arc sin, Ī etc illustrate the multiplicative version of convexity when restricted to appropriate subintervals of (0, ā) .As a consequence, we are not only able to improve on a number of classical elementary inequalities but also to discover new ones.
In this paper, we introduce a new approach on fractional integration, which generalizes the Riemann-Liouville fractional integral.We prove some properties for this new approach.We also establish some new integral inequalities ā¦
In this paper, we introduce a new approach on fractional integration, which generalizes the Riemann-Liouville fractional integral.We prove some properties for this new approach.We also establish some new integral inequalities using this new fractional integration.
In this paper, we set the main concepts for geometric (multiplicative) fractional calculus. We define Caputo, Riemann and Letnikov multiplicative fractional derivatives and multiplicative fractional integrals and study some of ā¦
In this paper, we set the main concepts for geometric (multiplicative) fractional calculus. We define Caputo, Riemann and Letnikov multiplicative fractional derivatives and multiplicative fractional integrals and study some of their properties. Finally, the multiplicative analogue of the local conformable fractional derivative and integral is studied.
The aim of this paper is to introduce a new extension of convexity called Ļ-convexity. We show that the class of Ļ-convex functions includes several other classes of convex functions. ā¦
The aim of this paper is to introduce a new extension of convexity called Ļ-convexity. We show that the class of Ļ-convex functions includes several other classes of convex functions. Some new integral inequalities of HermiteāHadamard type are established to illustrate the applications of Ļ-convex functions.
In this article, we define a new fractional technique which is known as generalized proportional fractional (GPF) integral in the sense of another function ĪØ . The authors prove several ā¦
In this article, we define a new fractional technique which is known as generalized proportional fractional (GPF) integral in the sense of another function ĪØ . The authors prove several inequalities for newly defined GPF-integral with respect to another function ĪØ . Our consequences will give noted outcomes for a suitable variation to the GPF-integral in the sense of another function ĪØ and the proportionality index Ļ . Furthermore, we present the application of the novel operator with several integral inequalities. A few new properties are exhibited, and the numerical approximation of these new operators is introduced with certain utilities to real-world problems.
In this paper, some integral inequalities of Hermite-Hadamard type for multiplicatively preinvex functions are established. Also, new inequalities involving multiplicative integrals are obtained for product and quotient of preinvex and ā¦
In this paper, some integral inequalities of Hermite-Hadamard type for multiplicatively preinvex functions are established. Also, new inequalities involving multiplicative integrals are obtained for product and quotient of preinvex and multiplicatively preinvex functions.
Abstract In this article, more general types of fractional proportional integrals and derivatives are proposed. Some properties of these operators are discussed.
Abstract In this article, more general types of fractional proportional integrals and derivatives are proposed. Some properties of these operators are discussed.
Integral inequality plays a critical role in both theoretical and applied mathematics fields. It is clear that inequalities aim to develop different mathematical methods (numerically or analytically) and to dedicate ā¦
Integral inequality plays a critical role in both theoretical and applied mathematics fields. It is clear that inequalities aim to develop different mathematical methods (numerically or analytically) and to dedicate the convergence and stability of the methods. Unfortunately, mathematical methods are useless if the method is not convergent or stable. Thus, there is a present day need for accurate inequalities in proving the existence and uniqueness of the mathematical methods. Convexity play a concrete role in the field of inequalities due to the behaviour of its definition. There is a strong relationship between convexity and symmetry. Which ever one we work on, we can apply to the other one due to the strong correlation produced between them especially in recent few years. In this article, we first introduced the notion of Ī» -incomplete gamma function. Using the new notation, we established a few inequalities of the HermiteāHadamard (HH) type involved the tempered fractional integrals for the convex functions which cover the previously published result such as Riemann integrals, RiemannāLiouville fractional integrals. Finally, three example are presented to demonstrate the application of our obtained inequalities on modified Bessel functions and q-digamma function.
The primary goal of this study is to define the weighted fractional operators on some spaces. We first prove that the weighted integrals are bounded in certain spaces. Afterwards, we ā¦
The primary goal of this study is to define the weighted fractional operators on some spaces. We first prove that the weighted integrals are bounded in certain spaces. Afterwards, we discuss the weighted fractional derivatives defined on absolute continuous-like spaces. At the end, we present a modified Laplace transform that can be applied perfectly to such operators.
In this study, we first establish two Hermite-Hadamard type inequality for multiplicative (geometric) Riemann-Liouville fractional integrals.Then, by using some properties of multiplicative convex function, we give some new inequalities involving ā¦
In this study, we first establish two Hermite-Hadamard type inequality for multiplicative (geometric) Riemann-Liouville fractional integrals.Then, by using some properties of multiplicative convex function, we give some new inequalities involving multiplicative fractional integrals.
Abstract The main aim this work is to give HermiteāHadamard inequalities for convex functions via generalized proportional fractional integrals. We also obtained extensions of HermiteāHadamard type inequalities for generalized proportional ā¦
Abstract The main aim this work is to give HermiteāHadamard inequalities for convex functions via generalized proportional fractional integrals. We also obtained extensions of HermiteāHadamard type inequalities for generalized proportional fractional integrals.
In this paper, we firstly obtain two identities for multiplicative differentiable functions. Then by using these identities, we establish Ostrowski and Simpson type inequalities for multiplicative integrals. At the end ā¦
In this paper, we firstly obtain two identities for multiplicative differentiable functions. Then by using these identities, we establish Ostrowski and Simpson type inequalities for multiplicative integrals. At the end we give detail applications of our main results
<abstract> In this paper, we propose a new framework of weighted generalized proportional fractional integral operator with respect to a monotone function $ \Psi, $ we develop novel modifications of ā¦
<abstract> In this paper, we propose a new framework of weighted generalized proportional fractional integral operator with respect to a monotone function $ \Psi, $ we develop novel modifications of the aforesaid operator. Moreover, contemplating the so-called operator, we determine several notable weighted Chebyshev and GrĆ¼ss type inequalities with respect to increasing, positive and monotone functions $ \Psi $ by employing traditional and forthright inequalities. Furthermore, we demonstrate the applications of the new operator with numerous integral inequalities by inducing assumptions on $ \omega $ and $ \Psi $ verified the superiority of the suggested scheme in terms of efficiency. Additionally, our consequences have a potential association with the previous results. The computations of the proposed scheme show that the approach is straightforward to apply and computationally very user-friendly and accurate. </abstract>
In this article, we establish the weighted (k,s)-Riemann-Liouville fractional integral and differential operators. Some certain properties of the operators and the weighted generalized Laplace transform of the new operators are ā¦
In this article, we establish the weighted (k,s)-Riemann-Liouville fractional integral and differential operators. Some certain properties of the operators and the weighted generalized Laplace transform of the new operators are part of the paper. The article consists of Chebyshev-type inequalities involving a weighted fractional integral. We propose an integro-differential kinetic equation using the novel fractional operators and find its solution by applying weighted generalized Laplace transforms.
<abstract><p>In this paper, we have established some new HermiteāHadamardāMercer type of inequalities by using $ {\kappa} $āRiemannāLiouville fractional integrals. Moreover, we have derived two new integral identities as auxiliary results. ā¦
<abstract><p>In this paper, we have established some new HermiteāHadamardāMercer type of inequalities by using $ {\kappa} $āRiemannāLiouville fractional integrals. Moreover, we have derived two new integral identities as auxiliary results. From the applied identities as auxiliary results, we have obtained some new variants of HermiteāHadamardāMercer type via $ {\kappa} $āRiemannāLiouville fractional integrals. Several special cases are deduced in detail and some know results are recaptured as well. In order to illustrate the efficiency of our main results, some applications regarding special means of positive real numbers and error estimations for the trapezoidal quadrature formula are provided as well.</p></abstract>
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 ā¦
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 HermiteāHadamard-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.
<abstract><p>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 ā¦
<abstract><p>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.</p></abstract>
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 ā¦
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.
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 ā¦
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.
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 ā¦
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 HermiteāHadamard 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 In this work, the notion of a multiplicative harmonic convex function is examined, and HermiteāHadamard inequalities for this class of functions are established. Many inequalities of HermiteāHadamard type are ā¦
Abstract In this work, the notion of a multiplicative harmonic convex function is examined, and HermiteāHadamard inequalities for this class of functions are established. Many inequalities of HermiteāHadamard 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.
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 ā¦
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.
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 ā¦
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 HermiteāHadamard type in the fractional domain. First, we propose the idea of generalized interval-valued convexity with respect to the continuous monotonic functions ā, bifunction Ī¶, and based on the containment ordering relation, which is termed as (ā,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 ā and h. Moreover, we use the notion of (ā,h)-pre-invexity and AtanganaāBaleanu (AB) fractional operators to develop some fresh fractional variants of the HermiteāHadamard (HH), Pachpatte, and HermiteāHadamardāFejer (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.
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 ā¦
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.
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 ā¦
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.
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. ā¦
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.
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 ā¦
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 Ī³, 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 RiemannāLiouville stochastic fractional operators and our proposed class, we prove parametric fractional variants of Jensenās inequality, HermiteāHadamardās inequality, Fejerās inequality, and product HermiteāHadamardās like inequality. We establish an interesting relation between means by means of HermiteāHadamardās 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 RiemannāLiouville fractional operators in different scenarios for a larger space of processes.
The relation between fractional calculus and convexity significantly impacts the development of the theory of integral inequalities. In this paper, we explore the reverse of Minkowski and Hƶlderās inequality, unified ā¦
The relation between fractional calculus and convexity significantly impacts the development of the theory of integral inequalities. In this paper, we explore the reverse of Minkowski and Hƶlderās inequality, unified Jensenās inequality, and HermiteāHadamard (H-H)-like inequalities using fractional calculus and a generic class of interval-valued convexity. We introduce the concept of I.V-(ā,ā) generic class of convexity, which unifies several existing definitions of convexity. By utilizing RiemannāLiouville (R-L) fractional operators and I.V-(ā,ā) convexity to derive new improvements of the H-H- and Fejer and Pachpatte-like inequalities. Our results are quite unified; by substituting the different values of parameters, we obtain a blend of new and existing inequalities. These results are fruitful for establishing bounds for I.V R-L integral operators. Furthermore, we discuss various implications of our findings, along with numerical examples and simulations to enhance the reliability of our results.