Preprint Article Version 1 This version is not peer-reviewed

# Ostrowski and Trapezoid Type Inequalities for the Generalized k-g-Fractional Integrals of Functions with Bounded Variation

Version 1 : Received: 6 December 2017 / Approved: 6 December 2017 / Online: 6 December 2017 (08:32:00 CET)

How to cite: Dragomir, S.S. Ostrowski and Trapezoid Type Inequalities for the Generalized k-g-Fractional Integrals of Functions with Bounded Variation. Preprints 2017, 2017120034 (doi: 10.20944/preprints201712.0034.v1). Dragomir, S.S. Ostrowski and Trapezoid Type Inequalities for the Generalized k-g-Fractional Integrals of Functions with Bounded Variation. Preprints 2017, 2017120034 (doi: 10.20944/preprints201712.0034.v1).

## Abstract

Let g be a strictly increasing function on $\left(a,b\right),$ having a continuous derivative g on $\left(a,b\right).$ For the Lebesgue integrable function $f:\left(a,b\right)\to \mathbb{C}$ , we define the k-g-left-sided fractional integral of f by ${S}_{k,g,a+}f\left(x\right)={\int }_{a}^{x}k\left(g\left(x\right)-g\left(t\right)\right){g}^{\prime }\left(t\right)f\left(t\right)dt,\phantom{\rule{4.pt}{0ex}}x\in \left(a,b\right]$ and the k-g-right-sided fractional integral of f by ${S}_{k,g,b-}f\left(x\right)={\int }_{x}^{b}k\left(g\left(t\right)-g\left(x\right)\right){g}^{\prime }\left(t\right)f\left(t\right)dt,\phantom{\rule{4.pt}{0ex}}x\in \left[a,b\right),$ where the kernel k is defined either on $\left(0,\infty \right)$ or on $\left[0,\infty \right)$ with complex values and integrable on any finite subinterval. In this paper we establish some Ostrowski and trapezoid type inequalities for the k-g-fractional integrals of functions of bounded variation. Applications for mid-point and trapezoid inequalities are provided as well. Some examples for a general exponential fractional integral are also given.

## Subject Areas

generalized Riemann-Liouville fractional integrals; Hadamard fractional integrals; functions of bounded variation; Ostrowski type inequalities; Trapezoid inequalities