- Advances in the Theory of Nonlinear Analysis and its Application
- Vol: 6 Issue: 4
- Ostrowski type inequalities via exponentially $s$-convexity on time scales
Ostrowski type inequalities via exponentially $s$-convexity on time scales
Authors : Svetlin GEORGİEV, Vahid DARVİSH, Eze NWAEZE
Pages : 502-512
Doi:10.31197/atnaa.1021333
View : 5 | Download : 1
Publication Date : 2022-12-30
Article Type : Research
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