Some fractional proportional integral inequalities

Abstract

In the last few years, various researchers studied the so-called conformable integrals and derivatives. Based on that notion some authors used modified conformable derivatives (proportional derivatives) to generate nonlocal fractional integrals and derivatives, called fractional proportional integrals and derivatives, which contain exponential functions in their kernels. Our aim in this paper is to establish some new integral inequalities by utilizing the fractional proportional-integral operators. In fact, certain new classes of integral inequalities for a class of n (n ∈ N) positive continuous and decreasing functions on [a, b] are presented. The inequalities presented in this paper are more general than the existing classical inequalities.

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

“…Rahman et al [43] established the Minkowski inequality and other types of inequalities in the frame of the proportional fractional integrals. In [44], Rahman et al discussed some specific new types of integral inequalities for a class of n (n ∈ N) positive continuous and decreasing functions on [r, b]. Rahman et al [45] defined the generalized proportional Hadamard fractional integrals and established certain new integral inequalities for convex functions.…”

Tempered Fractional Integral Inequalities for Convex Functions

“…Dahmani [47] presented some classes of fractional integral inequalities by considering a family of n positive functions. Certainly, remarkable inequalities such as Hermite-Hadamard type [48], Chebyshev type [49][50][51], inequalities via generalized conformable integrals [52], Grüss type [53,54], fractional proportional inequalities and inequalities for convex functions [55], Hadamard proportional fractional integrals [56], bounds of proportional integrals with applications [57], inequalities for the weighted and the extended Chebyshev functionals [58], certain new inequalities for a class of n(n ∈ N) positive continuous and decreasing functions [59] and certain generalized fractional inequalities [60] are recently presented by utilizing several different kinds of fractional calculus approaches.…”

On Some Fractional Integral Inequalities of Hermite-Hadamard’s Type through Convexity

“…Many researchers have extended the Hermite-Hadamard's inequality, to different forms using the classical convex function. For further details involving Hermite-Hadamard's type inequality on different concept of convex function and generalizations, the interested reader is referred to [2][3][4][5][6][7][8][9][10][11][12] and references therein.…”