New time scale generalizations of the Ostrowski-Gr\xfcss type inequality for k points

Eze R Nwaeze,Seth Kermausuor,Ana M Tameru

doi:10.1186/s13660-017-1525-6

Eze R Nwaeze, Seth Kermausuor + Show 1 more

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https://doi.org/10.1186/s13660-017-1525-6

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Abstract

Two Ostrowski-Grüss type inequalities for k points with a parameter lambdain[0, 1] are hereby presented. The first generalizes a recent result due to Nwaeze and Tameru, and the second extends the result of Liu et al. to k points. Many new interesting inequalities can be derived as special cases of our results by considering different values of λ and kinmathbb{N}. In addition, we apply our results to the continuous, discrete, and quantum time scales to obtain several novel inequalities in this direction.

Highlights

In, Dragomir and Wang [ ] obtained the following inequality which is today known as the Ostrowski-Grüss inequality.Theorem If f : [a, b] → R is differentiable on [a, b] and γ ≤ f (x) ≤ for all x ∈ [a, b] for some constants γ, ∈ R, f (x) –b f (b) – f (a) a + b f (t) dt – x–≤ (b – a)( – γ ) ( )b–a a b–a for all x ∈ [a, b].With the introduction of the theory of time scales, Tuna and Daghan [ ] obtained the following time scale version of the Ostrowski-Grüss type inequality
We present a brief overview of the theory of time scales
We introduce the sets Tk, Tk, and Tkk, which are derived from the time scale T as follows: if T has a left-scattered maximum t, Tk = T \ {t }, otherwise Tk = T

Summary

Introduction

In , Dragomir and Wang [ ] (see [ , ] for related results) obtained the following inequality which is today known as the Ostrowski-Grüss inequality. With the introduction of the theory of time scales (see Section ), Tuna and Daghan [ ] obtained the following time scale version of the Ostrowski-Grüss type inequality. Nwaeze and Tameru [ ] proved the following generalization of Theorem to k points. In Section , we recall some definitions and results of the time scale theory. Points that are right-scattered and left-scattered at the same time are called isolated. We introduce the sets Tk, Tk, and Tkk, which are derived from the time scale T as follows: if T has a left-scattered maximum t , Tk = T \ {t }, otherwise Tk = T. 3.1 Generalized Ostrowski-Grüss type inequality with a parameter I We state and prove our first result. We apply the items of Theorem , where applicable, to get b

From assumption

Conclusion