Motivated by some recent investigations about the sharpness of Jensen inequality, this paper deals with converse inequality. These results are then used for deriving new inequalities different types generalized f-divergences. As divergences measure differences between probability distributions, these applied on Zipf–Mandelbrot law as a special kind distribution.

Zipf’s law in its basic incarnation is an empirical probability distribution governing the frequency of usage of words in a language. As Terence Tao recently remarked, it still lacks a convincing and satisfactory mathematical explanation. In this paper I suggest that at least in certain situations, Zipf’s law can be explained as a special case of the a priori distribution introduced and studied...

We study the gross domestic product (GDP) per capita together with the market capitalization (MCAP) per capita as two indicators of the effect of globalization. We find that g, the GDP per capita, as a function of m, the MCAP per capita, follows a power law with average exponent close to 1/3. In addition, the Zipf ranking approach confirms that the m for countries with initially lower values of...

Usually, the studies of distributions of city populations have been reduced to power laws. In such analyses, a common practice is to consider cities with more than one hundred thousand inhabitants. Here, we argue that the distribution of cities for all ranges of populations can be well described by using a q-exponential distribution. This function, which reproduces the Zipf-Mandelbrot law, is r...

The pioneering research of G. K. Zipf on the relationship between word frequency and other word features led to the formulation of various linguistic laws. Here we focus on a couple of them: the meaning-frequency law, i.e. the tendency of more frequent words to be more polysemous, and the law of abbreviation, i.e. the tendency of more frequent words to be shorter. Here we evaluate the robustnes...

The derivation of the maximum entropy distribution of particles in boxes yields two kinds of distributions: a "bell-like" distribution and a long-tail distribution. The first one is obtained when the ratio between particles and boxes is low, and the second one when the ratio is high. The obtained long tail distribution yields correctly the empirical Zipf law, Pareto's 20:80 rule and Benford's l...