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

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

*g*be a strictly increasing function on $\left(a,b\right),$ having a continuous derivative

*g*

*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 ${S}_{k,g,b-}f\left(x\right)={\int}_{x}^{b}k\left(g\left(t\right)-g\left(x\right)\right){g}^{\prime}\left(t\right)f\left(t\right)dt,\phantom{\rule{4.pt}{0ex}}x\in [a,b),$ 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.

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