Majorization, Csiszár divergence and Zipf-Mandelbrot law

Abstract

Abstract In this paper we show how the Shannon entropy is connected to the theory of majorization. They are both linked to the measure of disorder in a system. However, the theory of majorization usually gives stronger criteria than the entropic inequalities. We give some generalized results for majorization inequality using Csiszár f-divergence. This divergence, applied to some special convex functions, reduces the results for majorization inequality in the form of Shannon entropy and the Kullback-Leibler divergence. We give several applications by using the Zipf-Mandelbrot law.

Keywords

• majorization inequality
• Csiszár f-divergence
• Shannon entropy
• Kullback-Leibler divergence
• Zipf-Mandelbrot law