Further inequalities for the generalized k-g-fractional integrals of functions with bounded variation

Year 2020, Volume: 69 Issue: 1, 49 - 72, 30.06.2020

Sever Dragomir

https://doi.org/10.31801/cfsuasmas.542665

Abstract

Let g be a strictly increasing function on (a,b), having a continuous derivative g′ on (a,b). For the Lebesgue integrable function f:(a,b)→C, we define the k-g-left-sided fractional integral of f by

S_{k,g,a+}f(x)=∫_{a}^{x}k(g(x)-g(t))g′(t)f(t)dt, x∈(a,b]

and the k-g-right-sided fractional integral of f by

S_{k,g,b-}f(x)=∫_{x}^{b}k(g(t)-g(x))g′(t)f(t)dt, x∈[a,b),

where the kernel k is defined either on (0,∞) or on [0,∞) with complex values and integrable on any finite subinterval.

In this paper we establish some new inequalities for the k-g-fractional integrals of functions of bounded variation.Examples for the generalized left- and right-sided Riemann-Liouville fractional integrals of a function f with respect to another function g and a general exponential fractional integral are also provided.

Keywords

Generalized Riemann-Liouville fractional integrals, Hadamard fractional integrals, Functions of bounded variation, Ostrowski type inequalities

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