Bounds of Generalized Proportional Fractional Integrals in General Form via Convex Functions and Their Applications

Abstract

In this paper, our objective is to apply a new approach to establish bounds of sums of left and right proportional fractional integrals of a general type and obtain some related inequalities. From the obtained results, we deduce some new inequalities for classical generalized proportional fractional integrals as corollaries. These inequalities have a connection with some known and existing inequalities which are mentioned in the literature. In addition, some applications of the main results are presented.

On generalized fractional integral inequalities for the monotone weighted Chebyshev functionals

“…Rahman et al [54] recently investigated Grüss-type inequalities for generalized k-fractional conformable integrals. Minkowski's inequalities, fractional Hadamard proportional integral inequalities, and fractional proportional inequalities for convex functions by employing fractional proportional integrals can be found in [46][47][48][49][50][51][52][53]. In addition, various applications of fractional calculus can be found in [1, 17, 18, 28-32, 62, 64].…”

More properties of the proportional fractional integrals and derivatives of a function with respect to another function

“…Later, the authors in [24] presented a new type of fractional operators generated from the above-mentioned modified conformable derivatives. In addition, more generalized forms of these fractional operators were put forward in [25], and it turned out that some of these operators coincided with some operators mentioned before in [26][27][28].…”

Certain Hadamard Proportional Fractional Integral Inequalities

“…They established certain inequalities for convex functions by employing Hadamard proportional fractional integrals. In [49], the authors defined bounds of generalized proportional fractional integral operators for convex functions and their applications. Motivated by the above, here we presented certain inequalities by employing Hadamard proportional fractional integrals.…”