Certain Hadamard Proportional Fractional Integral Inequalities

Proportional Caputo Fractional Differential Inclusions in Banach Spaces

In this work, we introduce the notion of a (weak) proportional Caputo fractional derivative of order α∈(0,1) for a continuous (locally integrable) function u:[0,∞)→E, where E is a complex Banach space. In our definition, we do not require that the function u(·) is continuously differentiable, which enables us to consider the wellposedness of the corresponding fractional relaxation problems in a much better theoretical way. More precisely, we systematically investigate several new classes of (degenerate) fractional solution operator families connected with the use of this type of fractional derivatives, obeying the multivalued linear approach to the abstract Volterra integro-differential inclusions. The quasi-periodic properties of the proportional fractional integrals as well as the existence and uniqueness of almost periodic-type solutions for various classes of proportional Caputo fractional differential inclusions in Banach spaces are also considered.

On Fractional Inequalities Using Generalized Proportional Hadamard Fractional Integral Operator

The main objective of this paper is to use the generalized proportional Hadamard fractional integral operator to establish some new fractional integral inequalities for extended Chebyshev functionals. In addition, we investigate some fractional integral inequalities for positive continuous functions by employing a generalized proportional Hadamard fractional integral operator. The findings of this study are theoretical but have the potential to help solve additional practical problems in mathematical physics, statistics, and approximation theory.

Reverse Minkowski Inequalities Pertaining to New Weighted Generalized Fractional Integral Operators

In this paper, we obtain reverse Minkowski inequalities pertaining to new weighted generalized fractional integral operators. Moreover, we derive several important special cases for suitable choices of functions. In order to demonstrate the efficiency of our main results, we offer many concrete examples as applications.

Certain Inequalities Pertaining to Some New Generalized Fractional Integral Operators

In this paper, we introduce the generalized left-side and right-side fractional integral operators with a certain modified ML kernel. We investigate the Chebyshev inequality via this general family of fractional integral operators. Moreover, we derive new results of this type of inequalities for finite products of functions. In addition, we establish an estimate for the Chebyshev functional by using the new fractional integral operators. From our above-mentioned results, we find similar inequalities for some specialized fractional integrals keeping some of the earlier results in view. Furthermore, two important results and some interesting consequences for convex functions in the framework of the defined class of generalized fractional integral operators are established. Finally, two basic examples demonstrated the significance of our results.

On the Fractional Maximal Delta Integral Type Inequalities on Time Scales

Time scales have been the target of work of many mathematicians for more than a quarter century. Some of these studies are of inequalities and dynamic integrals. Inequalities and fractional maximal integrals have an important place in these studies. For example, inequalities and integrals contributed to the solution of many problems in various branches of science. In this paper, we will use fractional maximal integrals to establish integral inequalities on time scales. Moreover, our findings show that inequality is valid for discrete and continuous conditions.