ON SOME NEW HERMITE–HADAMARD–MERCER-TYPE INEQUALITIES THROUGH SEPARABLE SEQUENCES AND CAPUTO FRACTIONAL OPERATORS

In this paper, we primarily deal with approximately monotone and convex sequences. We start by showing that any sequence can be expressed as the difference of two nondecreasing sequences. One of these two monotone sequences acts as the majorant of the original sequence, while the other possesses non-negativity. Another result establishes that an approximately monotone (increasing) sequence can be closely approximated by a nondecreasing sequence. A similar assertion can be made for an approximately convex sequence. A sequence $\big<u_n\big>_{n=0}^{\infty}$ is called approximately convex (or $\varepsilon$-convex) if the following inequality holds under the mentioned assumptions: $$u_{i}-u_{i-1}\leq u_{j}-u_{j-1}+\varepsilon,\quad\quad\mbox{ where } i,j\in\mathbb{N},\quad \mbox{ with } i<j.$$ We prove that an approximately convex sequence can be written as the algebraic difference of two specific types of sequences. The initial sequence contains sequential convexity property, while the other sequence possesses the Lipschitz property. Moreover, we introduce an operator $\mathscr{T}$, which is termed as a twisting operator. In a compact interval $I(\subseteq\mathbb{R})$, we characterize the convex function with this newly introduced operator. In addition to various results on sequence decomposition and the study of the $\mathscr{T}$-operator, a characterization regarding non-negative sequential convexity, a fractional inequality, the implication of the $\mathscr{T}$ operator on different types of functions, the relationship between a convex function and a convex sequence are also included. Motivation, previous research in this direction, various applications and linkage with some other branches of mathematics are discussed in the Introduction.

Fractional derivative and integral operators are often employed to present new generalizations of mathematical inequalities. The introduction of new fractional operators has prompted another direction in different branches of mathematics and applied sciences. First, we investigate and prove new fractional equality. Considering this equality as the auxiliary result, we attain some estimations of a Hermite–Hadamard type inequality involving s-preinvex, s-Godunova–Levin preinvex, and prequasi invex functions. In addition, we investigate a fractional order Hadamard–Fejér inequality and some of its refinements pertaining to h-preinvexity via a non-conformable fractional integral operator. Finally, we present a Pachpatte type inequality for the product of two preinvex functions. The findings as well as the special cases presented in this research are new and applications of our main results.

<abstract><p>A new approach is used to investigate the analytical solutions of the mathematical fractional Casson fluid model that is described by the Constant Proportional Caputo fractional operator having non-local and singular kernel near an infinitely vertical plate. The phenomenon has been expressed in terms of partial differential equations, and the governing equations were then transformed in non-dimensional form. For the sake of generalized memory effects, a new mathematical fractional model is formulated based on the newly introduced Constant Proportional Caputo fractional derivative operator. This fractional model has been solved analytically, and exact solutions for dimensionless velocity, concentration and energy equations are calculated in terms of Mittag-Leffler functions by employing the Laplace transformation method. For the physical significance of various system parameters such as $ \alpha $, $ \beta $, $ Pr $, $ Gr $, $ Gm $, $ Sc $ on velocity, temperature and concentration profiles, different graphs are demonstrated by Mathcad software. The Constant Proportional Caputo fractional parameter exhibited a retardation effect on momentum and energy profile, but it is visualized that for small values of Casson fluid parameter, the velocity profile is higher. Furthermore, to validated the acquired solutions, some limiting models such as the ordinary Newtonian model are recovered from the fractionalized model. Moreover, the graphical representations of the analytical solutions illustrated the main results of the present work. Also, from the literature, it is observed that to deriving analytical results from fractional fluid models developed by the various fractional operators is difficult, and this article contributes to answering the open problem of obtaining analytical solutions for the fractionalized fluid models.</p></abstract>

The main aim of this paper is to give the definitions of Caputo fractional derivative operators and show their use in the special function theory. For this purpose, we introduce new types of incomplete hypergeometric functions and obtain their integral representations. Furthermore, we define incomplete Caputo fractional derivative operators and show that the images of some elementary functions under the action of incomplete Caputo fractional operators give a new type of incomplete hypergeometric functions. This definition helps us to obtain linear and bilinear generating relations for the new type incomplete Gauss hypergeometric functions.

In this article, we investigate the solution of the fractional multidimensional Navier–Stokes equation based on the Caputo fractional derivative operator. The behavior of the solution regarding the Navier–Stokes equation system using the Sumudu transform approach is discussed analytically and further discussed graphically.

This paper is concerned to establish an advanced form of the well-known Hermite-Hadamard (HH) inequality for recently-defined Generalized Conformable (GC) fractional operators. This form of the HH inequality combines various versions (new and old) of this inequality, containing operators of the types Katugampula, Hadamard, Riemann-Liouville, conformable and Riemann, into a single form. Moreover, a novel identity containing the new GC fractional integral operators is proved. By using this identity, a bound for the absolute of the difference between the two rightmost terms in the newly-established Hermite-Hadamard inequality is obtained. Also, some relations of our results with the already existing results are presented. Conclusion and future works are presented in the last section.

Modified Atangana-Baleanu fractional operators involving generalized Mittag-Leffler function

In studying boundary value problems and coupled systems of fractional order in (1,2], involving Hilfer fractional derivative operators, a zero initial condition is necessary. The consequence of this fact is that boundary value problems and coupled systems of fractional order with non-zero initial conditions cannot be studied. For example, such boundary value problems and coupled systems of fractional order are those including separated, non-separated, or periodic boundary conditions. In this paper, we propose a method for studying a coupled system of fractional order in (1,2], involving fractional derivative operators of Hilfer and Caputo with non-separated boundary conditions. More precisely, a sequential coupled system of fractional differential equations including Hilfer and Caputo fractional derivative operators and non-separated boundary conditions is studied in the present paper. As explained in the concluding section, the opposite combination of Caputo and Hilfer fractional derivative operators requires zero initial conditions. By using Banach’s fixed point theorem, the uniqueness of the solution is established, while by applying the Leray–Schauder alternative, the existence of solution is obtained. Numerical examples are constructed illustrating the main results.

In this paper, we conduct a research on a new version of the ( p,q ) -Hermite–Hadamard inequality for convex functions in the framework of postquantum calculus. Moreover, we derive several estimates for (p,q)-midpoint and (p,q)-trapezoidal inequalities for special ( p,q ) -differentiable functions by using the notions of left and right (p,q ) -derivatives. Our newly obtained inequalities are extensions of some existing inequalities in other studies. Lastly, we consider some mathematical examples for some (p,q)-functions to confirm the correctness of newly established results.

This study generalizes Hermite–Hadamard–Mercer type inequalities using Riemann–Liouville fractional integrals within the framework of multiplicative calculus. Multiplicative fractional integral identities are established for ∗differentiable convex functions, forming the basis for deriving trapezoidal-Mercer and midpoint-Mercer type inequalities in this context. Numerical examples and graphical analysis are provided to demonstrate the applicability and effectiveness of these inequalities, along with applications to special means of real numbers. For the first time, applications to quadrature formulas for midpoint and trapezoidal rules are presented within the framework of multiplicative calculus. This work underscores the versatility of the fractional operator in addressing problems involving noninteger order differentiation, offering refinements to classical inequalities. By extending these inequalities, the research aims to uncover new mathematical insights, properties, and relationships, contributing to developing advanced mathematical tools applicable across various scientific disciplines.

The symmetric function class interacts heavily with other types of functions. One of these is the convex function class, which is strongly related to symmetry theory. In this study, we define a novel class of convex mappings on planes using a fuzzy inclusion relation, known as coordinated up and down convex fuzzy-number-valued mapping. Several new definitions are introduced by placing some moderate restrictions on the notion of coordinated up and down convex fuzzy-number-valued mapping. Other uncommon examples are also described using these definitions, which can be viewed as applications of the new outcomes. Moreover, Hermite–Hadamard–Fejér inequalities are acquired via fuzzy double Aumann integrals, and the validation of these outcomes is discussed with the help of nontrivial examples and suitable choices of coordinated up and down convex fuzzy-number-valued mappings.

Korovkin type theorems and approximate Hermite–Hadamard inequalities

In this paper, we present a set of Newton‐type inequalities for n‐times differentiable convex functions using the Caputo fractional operator, extending classical results into the fractional calculus domain. Our exploration also includes the derivation of Newton‐type inequalities for various classes of functions by employing the Caputo fractional operator, thereby broadening the scope of these inequalities beyond convexity. In addition, we establish several fractional Newton‐type inequalities by using bounded functions in conjunction with fractional integrals. Furthermore, we develop specific fractional Newton‐type inequalities tailored to Lipschitzian functions. Moreover, the paper emphasizes the significance of fractional calculus in refining classical inequalities and demonstrates how the Caputo fractional operator provides a more generalized framework for addressing problems involving non‐integer order differentiation. The inclusion of bounded and Lipschitzian functions introduces additional layers of complexity, allowing for a more comprehensive analysis of function behaviors under fractional operations.

For k-Riemann–Liouville fractional integral operators, the Hermite–Hadamard inequality is already well-known in the literature. In this regard, this paper presents the Hermite–Hadamard inequalities for k-Riemann–Liouville fractional integral operators by using a novel method based on Green’s function. Additionally, applying these identities to the convex and monotone functions, new Hermite–Hadamard type inequalities are established. Furthermore, a different form of the Hermite–Hadamard inequality is also obtained by using this novel approach. In conclusion, we believe that the approach presented in this paper will inspire more research in this area.

The fractional input stability of the electrical circuit equations described by the fractional derivative operators has been investigated. The Riemann-Liouville and the Caputo fractional derivative operators have been used. The analytical solutions of the electrical circuit equations have been developed. The Laplace transforms of the Riemann-Liouville, and the Caputo fractional derivative operators have been used. The graphical representations of the analytical solutions of the electrical circuit equations have been presented. The converging-input converging-state property of the electrical RL, RC and LC circuit equations described by the Caputo fractional derivative, and the global asymptotic stability property of the unforced electrical circuit equations have been illustrated.