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

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Publication Date : 2022-12-30

Article Type : Research Paper

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 finsert ignore into journalissuearticles values(t);\\Delta t\\\\ &\\leq \\frac{finsert ignore into journalissuearticles values(a);}{e_{\\beta}insert ignore into journalissuearticles values(a, x_0); insert ignore into journalissuearticles values(b-a);^{2s}}insert ignore into journalissuearticles values(h_2insert ignore into journalissuearticles values(a, b););^s+\\frac{finsert ignore into journalissuearticles values(b);}{e_{\\beta}insert ignore into journalissuearticles values(b, x_0); insert ignore into journalissuearticles values(b-a);^{2s}}insert ignore into journalissuearticles values(h_2insert ignore into journalissuearticles values(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