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

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

Kaynakça

1. [1] R. P. Agarwal and M. Bohner. Basic calculus on time scales and some of its applications. Results Math., 35(1-2):3-22, 1999.
2. [2] R. Agarwal, M. Bohner, and A. Peterson. Inequalities on time scales: A survey. Math. Inequal. Appl., 4(4):535-557, 2001.
3. [3] M. Bohner and A. Peterson. Dynamic Equations on Time Scales: An Introduction with Applications. Birkhauser, Boston, 2001.
4. [4] M. Bohner and T. Matthews. Ostrowski inequalities on time scales. JIPAM. J. Inequal. Pure Appl. Math., 9(1):8, 2008. Article 6.
5. [5] M. Bohner and S. G. Georgiev, Multivariable dynamic calculus on time scales. Springer, Cham, 2016.
6. [6] M. Bohner, R. A. C. Ferreira and D. F. M. Torres, Integral inequalities and their applications to the calculus of variations on time scales, Math. Inequal. Appl., 13(3):511-522, 2010.
7. [7] S. G. Georgiev, V. Darvish and T. A. Roushan, Some inequalities for exponentially convex functions on time scales, Mathematica Slovaca, 71 (4): 925-940, 2021.
8. [8] S. Hilger, Ein Maβkettenkalkül mit Anwendung Zentrumsmannigfaltigkeiten. PhD thesis, Universitat Würzburg, 1988.

9. [9] N. Mehreen and M. Anwar. Ostrowski type inequalities via some exponentially convex functions with applications, AIMS Mathematics, 5(2) 1476-1483.
10. [10] N. Mehreen, M. Anwar, Hermite-Hadamard type inequalities for exponentially p-convex functions and exponentially s- convex functions in the second sense with applications. J Inequal Appl 2019, 92 (2019). https://doi.org/10.1186/s13660- 019-2047-1
11. [11] E. R. Nwaeze, Generalized weighted trapezoid and Grüss type inequalities on time scales, Aust. J. Math. Anal. Appl., 14(1):Art. 4, 13, 2017.
12. [12] E. R. Nwaeze, Time scale versions of the Ostrowski?Grüss type inequality with a parameter function, J. Math. Inequal., 12, 531-543, 2018.