(k, ψ)-Proportional Fractional Integral Pólya–Szegö- and Grüss-Type Inequalities

Abstract

The purpose of this research was to discover a novel method to recover k-fractional integral inequalities of the Pólya–Szegö-type. We employ these generalized inequalities to investigate some new fractional integral inequalities of the Grüss-type. More precisely, we generalize the proportional fractional operators with respect to another strictly increasing continuous function ψ. Then, we state and prove some of its properties and special cases. With the help of this generalized operator, we investigate some Pólya–Szegö- and Grüss-type fractional integral inequalities. The functions used in this work are bounded by two positive functions to obtain Pólya–Szegö- and Grüss-type k-fractional integral inequalities in a new sense. Moreover, we discuss some new special cases of the Pólya–Szegö- and Grüss-type inequalities through this work.

Editorial for the Special Issue “Operators of Fractional Calculus and Their Multidisciplinary Applications”

“…The geographical distribution of the contributors to this Special Issue is remarkably widely-scattered. Their contributions (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]) originated in many different countries on every continent of the world. The subject matter of these nineteen publications (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]) deals extensively with such topics as fractional-order complex Ginzburg-Landau equations, fractional modeling for the treatment of cancer by using radiotherapy, fractional-order fuzzy complex-valued neural networks, the fractal-fractional Michaelis-Menten enzymatic reaction model, fractional-calculus operators involving the (p, q)-extended Bessel and Bessel-Wright functions, fractional-order diffusion-wave equations, Abel integral equations and their fractional-order analogues, nonlinear integro-differential equations, fractionalorder investigations of a number of celebrated integral inequalities, such as those that are popularly called the Pólya-Szegö inequality, the Grüss inequality, the Hermite-Hadamard inequality, and so on.…”

“…The so-called (k, Ψ)-proportional fractional-order integral inequalities of the Pólya-Szegö and Grüss types are investigated in [13]. Several fractional-order integral inequalities based upon some general families of fractional integrals are investigated by the authors in [14,15].…”

Chebyshev-Type Inequalities Involving ( k , ψ )-Proportional Fractional Integral Operators

“…Definition 3 (see [10]). Let α > 0,k > 0,ν ∈ ð0, 1: Let f be an integrable function and ψ be a continuous increasing function on ½a, b.…”

“…In particular, Jarad et al [8] has contributed significantly to the study of FC by introducing generalized proportional fractional operators through exponential kernel, while in [9], the authors extended this work and defined the proportional fractional operator of a function with respect to the another function. Furthermore, in [10],…”

“…Rashid et al [12] presented reverse Minkowski's type inequalities by using generalized proportional fractional operators. In [10], authors studied Pólya-Szegö-type inequalities and Grüss-type inequalities with the help of ðk, ψÞ-proportional fractional operators. In recent years, many researchers have been working in the direction of estimating the fractional version of various inequalities [13][14][15][16][17][18][19] and the references given there.…”