Some Novel Fractional Integral Inequalities over a New Class of Generalized Convex Function

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.

Some Midpoint Inequalities for η -Convex Function via Weighted Fractional Integrals

In this research, by using a weighted fractional integral, we establish a midpoint version of Hermite-Hadamrad Fejér type inequality for η -convex function on a specific interval. To confirm the validity, we considered some special cases of our results and relate them with already existing results. It can be observed that several existing results are special cases of our presented results.

Orthonormal piecewise Bernoulli functions: Application for optimal control problems generated using fractional integro-differential equations

In this study, the orthonormal piecewise Bernoulli functions are generated as a new kind of basis functions. An explicit matrix related to fractional integration of these functions is obtained. An efficient direct method is developed to solve a novel set of optimal control problems defined using a fractional integro-differential equation. The presented technique is based on the expressed basis functions and their fractional integral matrix together with the Gauss–Legendre integration method and the Lagrange multipliers algorithm. This approach converts the original problem into a mathematical programming one. Three examples are investigated numerically to verify the capability and reliability of the approach.

Riemann–Liouville Fractional Integral Inequalities for Generalized Pre-Invex Functions of Interval-Valued Settings Based upon Pseudo Order Relation

The concepts of convex and non-convex functions play a key role in the study of optimization. So, with the help of these ideas, some inequalities can also be established. Moreover, the principles of convexity and symmetry are inextricably linked. In the last two years, convexity and symmetry have emerged as a new field due to considerable association. In this paper, we study a new version of interval-valued functions (I-V·Fs), known as left and right χ-pre-invex interval-valued functions (LR-χ-pre-invex I-V·Fs). For this class of non-convex I-V·Fs, we derive numerous new dynamic inequalities interval Riemann–Liouville fractional integral operators. The applications of these repercussions are taken into account in a unique way. In addition, instructive instances are provided to aid our conclusions. Meanwhile, we’ll discuss a few specific examples that may be extrapolated from our primary findings.

Extension of Darbo’s fixed point theorem via shifting distance functions and its application

In this paper, we discuss solvability of infinite system of fractional integral equations (FIE) of mixed type. To achieve this goal, we first use shifting distance function to establish a new generalization of Darbo’s fixed point theorem, and then apply it to the FIEs to establish the existence of solution on tempered sequence space. Finally, we verify our results by considering a suitable example.

Mixed-Norm Amalgam Spaces and Their Predual

In this paper, we introduce mixed-norm amalgam spaces (Lp→,Ls→)α(Rn) and prove the boundedness of maximal function. Then, the dilation argument obtains the necessary and sufficient conditions of fractional integral operators’ boundedness. Furthermore, the strong estimates of linear commutators [b,Iγ] generated by b∈BMO(Rn) and Iγ on mixed-norm amalgam spaces (Lp→,Ls→)α(Rn) are established as well. In order to obtain the necessary conditions of fractional integral commutators’ boundedness, we introduce mixed-norm Wiener amalgam spaces (Lp→,Ls→)(Rn). We obtain the necessary and sufficient conditions of fractional integral commutators’ boundedness by the duality theory. The necessary conditions of fractional integral commutators’ boundedness are a new result even for the classical amalgam spaces. By the equivalent norm and the operators Str(p)(f)(x), we study the duality theory of mixed-norm amalgam spaces, which makes our proof easier. In particular, note that predual of the primal space is not obtained and the predual of the equivalent space does not mean the predual of the primal space.

On Hadamard Type Fractional Inequalities for Riemann–Liouville Integrals via a Generalized Convexity

In the literature of mathematical inequalities, convex functions of different kinds are used for the extension of classical Hadamard inequality. Fractional integral versions of the Hadamard inequality are also studied extensively by applying Riemann–Liouville fractional integrals. In this article, we define (α,h−m)-convex function with respect to a strictly monotone function that unifies several types of convexities defined in recent past. We establish fractional integral inequalities for this generalized convexity via Riemann–Liouville fractional integrals. The outcomes of this work contain compact formulas for fractional integral inequalities which generate results for different kinds of convex functions.

Generalized fractal Jensen and Jensen–Mercer inequalities for harmonic convex function with applications

In this paper, we present a generalized Jensen-type inequality for generalized harmonically convex function on the fractal sets, and a generalized Jensen–Mercer inequality involving local fractional integrals is obtained. Moreover, we establish some generalized Jensen–Mercer-type local fractional integral inequalities for harmonically convex function. Also, we obtain some generalized related results using these inequalities on the fractal space. Finally, we give applications of generalized means and probability density function.

New fractional identities, associated novel fractional inequalities with applications to means and error estimations for quadrature formulas

In this paper, the authors derive some new generalizations of fractional trapezium-like inequalities using the class of harmonic convex functions. Moreover, three new fractional integral identities are given, and on using them as auxiliary results some interesting integral inequalities are found. Finally, in order to show the efficiency of our main results, some applications to special means for different positive real numbers and error estimations for quadrature formulas are obtained.