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

Silvestru Sever Dragomir

doi:10.20944/preprints201712.0034.v1

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preprints.org > computer science and mathematics > analysis > doi: 10.20944/preprints201712.0034.v1

Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

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

Silvestru Sever Dragomir

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. https://doi.org/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. https://doi.org/10.20944/preprints201712.0034.v1

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MDPI and ACS Style

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

APA Style

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

Chicago/Turabian Style

Dragomir, S.S. 2017 "Ostrowski and Trapezoid Type Inequalities for the Generalized k-g-Fractional Integrals of Functions with Bounded Variation" Preprints. https://doi.org/10.20944/preprints201712.0034.v1

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) \to C

, we define the k-g-left-sided fractional integral of f by

S_{k, g, a +} f (x) = \int_{a}^{x} k (g (x) - g (t)) g^{'} (t) f (t) d t, x \in (a, b]

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

S_{k, g, b -} f (x) = \int_{x}^{b} k (g (t) - g (x)) g^{'} (t) f (t) d t, x \in [a, b),

where the kernel k is defined either on

(0, \infty)

or on

[0, \infty)

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.

Keywords

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

Subject

Computer Science and Mathematics, Analysis

Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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