Some generalized Riemann-Liouville k-fractional integral inequalities

Praveen Agarwal,Sotiris K Ntouyas,Jessada Tariboon

doi:10.1186/s13660-016-1067-3

Abstract

The focus of the present study is to prove some new Polya-Szego type integral inequalities involving the generalized Riemann-Liouville k-fractional integral operator. These inequalities are used then to establish some fractional integral inequalities of Chebyshev type.

Highlights

1 Introduction and motivation The celebrated functionals were introduced by the Chebyshev in his famous paper [ ] and were subsequently rediscovered in various inequalities by numerous authors, including Anastassiou [ ], Belarbi and Dahmani [ ], Dahmani et al [ ], Dragomir [ ], Kalla and Rao [ ], Lakshmikantham and Vatsala [ ], Ntouyas et al [ ], Ögünmez and Özkan [ ], Sudsutad et al [ ], Sulaiman [ ]; and, for very recent work, see Wang et al [ ]
We organize the paper as follows: in Section, we prove some generalized Pólya-Szegö type integral inequalities involving the generalized Riemann-Liouville k-fractional integral operators that we need to establish main theorems in the sequel and Section contains some Chebyshev type integral inequalities via generalized Riemann-Liouville k-fractional integral operators
2 Some Pólya-Szegö types inequalities we prove some Pólya-Szegö type integral inequalities for positive integrable functions involving the generalized Riemann-Liouville k-fractional integral operator ( . )

Summary

MN mn

Proof Let f and g be two positive integrable functions on [a, ∞). Let f and g be two positive integrable functions on [a, ∞), a ≥. Assume that there exist four positive integrable functions φ , φ , ψ , and ψ satisfying (H ). For t > a, k > , a ≥ , α > , β > , and r ∈ R \ {– }, the following inequality is true: k) Raβ,,kr{fg}(t). With ψ (t) = ψ (t) = g(t) = , we have (Raα,,kr{(φ + φ )f }(t)) Raα,,kr {φ φ }(t)

This implies that

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