Ostrowski type inequalities via exponentially $s$-convexity on time scales

Year 2022, Volume: 6 Issue: 4, 502 - 512, 30.12.2022

Svetlin Georgiev Vahid Darvish Eze Nwaeze

https://doi.org/10.31197/atnaa.1021333

Abstract

We introduce the concept of exponentially $s$-convexity in the second sense on a time scale interval. We prove among other things that if $f: [a, b]\to \mathbb{R}$ is an exponentially $s$-convex function, then
\begin{align*}
&\frac{1}{b-a}\int_a^b f(t)\Delta t\\
&\leq \frac{f(a)}{e_{\beta}(a, x_0) (b-a)^{2s}}(h_2(a, b))^s+\frac{f(b)}{e_{\beta}(b, x_0) (b-a)^{2s}}(h_2(b, a))^s,
\end{align*}
where $\beta$ is a positively regressive function. By considering special cases of our time scale, one can derive loads of interesting new inequalities. The results obtained herein are novel to best of our knowledge and they complement existing results in the literature.

Keywords

Ostrowski inequality, Time scales, Hölder's inequality, Exponentially s-convexity

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