Estimations of the Slater Gap via Convexity and Its Applications in Information Theory

Abstract

Convexity has played a prodigious role in various areas of science through its properties and behavior. Convexity has booked record developments in the field of mathematical inequalities in the recent few years. The Slater inequality is one of the inequalities which has been acquired with the help of convexity. In this note, we obtain some estimations for the Slater gap while dealing with the notion of convexity in an extensive manner. We acquire the deliberated estimations by utilizing the definition of convex function, Jensen’s inequality for concave functions, and triangular, power mean, and Hölder inequalities. We discuss several consequences of the main results in terms of inequalities for the power means. Moreover, by utilizing the main results, we give estimations for the Csiszár and Kullback–Leibler divergences, Shannon entropy, and the Bhattacharyya coefficient. Furthermore, we present some estimations for the Zipf–Mandelbrot entropy as additional applications of the acquired results. The perception and approaches adopted in this note may pretend more research in this direction.

Derivation of Bounds for Majorization Differences by a Novel Method and Its Applications in Information Theory

“…Remark 2. A similar estimate given on the right side of inequality (15) was also presented in inequality (9) as stated in the article [58] for the absolute Slater difference.…”

“…Remark 7. The estimate provided by Khan et al [58] in (13) for the absolute Slater difference will coincide with the estimate on the right side of (25) by just taking p = 1 in (13).…”

“…Remark 4. The estimate provided in (20) for the Slater difference is also acquired by Khan et al [58] in inequality (15) for the absolute Slater difference.…”

“…Remark 9. An estimate for the Slater difference given on the right side of (30) is also produced by Khan et al [58] in (19) for the absolute Slater difference.…”

“…Remark 11. Similarly, a type of estimates for the Slater difference given in (35) can be studied in (21), which is stated in the article [58] for the absolute Slater difference.…”