Abstract
A number of Chebyshev type inequalities involving various fractional integral operators have, recently, been presented. In this work, motivated essentially by the earlier works and their applications in diverse research subjects, we establish some new Polya-Szego inequality involving generalized Katugampola fractional integral operator and use them to prove some new fractional Chebyshev type inequalities which are extensions of the results in the paper: [On Polya-Szego and Chebyshev type inequalities involving the Riemann-Liouville fractional integral operators, J. Math. Inequal, 10(2) (2016)].
Highlights
Introduction and preliminariesChebyshev inequality, the reader is refer to [2,3,4,5,6].This article is based on the well known Chebyshev functional [1]: We need to introduce the Polya and Szegoinequality [7]: T (f, g) b =1 b−a f (x) g (x) dx a − 1 −a f (x) dx a g (x) dx, a a where f and g are two integrable functions which are synchronous on [a, b], i.e.(f (x) − f (y))(g(x) − g(y)) ≥ 0 for any x, y ∈ [a, b], the Chebyshev inequality states that T (f, g) ≥ 0
1 4 is best possible in in the sense it can not be replaced by a smaller constant
We recall some other preliminaries: We note that the beta function B(α, β) is defined by
Summary
The reader is refer to [2,3,4,5,6]. This article is based on the well known Chebyshev functional [1]: We need to introduce the Polya and Szegoinequality [7]:. A f (x) dx a g (x) dx , a a where f and g are two integrable functions which are synchronous on [a, b], i.e. (f (x) − f (y))(g(x) − g(y)) ≥ 0 for any x, y ∈ [a, b], the Chebyshev inequality states that T (f, g) ≥ 0.
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