Chebyshev-Type Inequalities Involving ($k,\psi$)-Proportional Fractional Integral Operators

Abstract

Expanding the analytical aspect of mathematics enables researchers to study more cosmic phenomena, especially with regard to the applied sciences related to fractional calculus. In the present paper, we establish some Chebyshev-type inequalities in the case synchronous functions. In order to achieve our goals, we use k , ψ -proportional fractional integral operators. Moreover, we present some special cases.

Chebyshev–Jensen-Type Inequalities Involving χ-Products and Their Applications in Probability Theory

“…The main contributions of this article are that we extended the special function Π in the Chebyshev inequalities (11) and (13) to the general function χ, and we extended the m = 2 in the Chebyshev inequalities (11) and (13) to the m ≥ 2.…”

“…Then, by ( 5) and ( 6), we have the following Corollary 1. Therefore, Theorem 1 is a generalization of the discrete Chebyshev inequality (11).…”

“…The classical Chebyshev inequalities [1][2][3][4][5][6][7][8][9][10][11][12][13] can be expressed as follows. Let X 1 , X 2 ∈ R n .…”

“…In [1], the authors points out that the classical Chebyshev's integral inequality is deeply connected with the study of positive dependence of random variables, which are monotone functions of a common random variable. The Chebyshev-type inequalities and their applications were investigated by many authors [1][2][3][4][5][6][7][8][9][10][11][12][13]. In [2], the authors established the Chebyshev-type inequalities involving the permanents of matrix as follows:…”