Adil Khan Muhammad, Al-Sahwi Zaid Mohammad, Chu Yu-Ming
College of Science, Hunan City University, Yiyang 413000, China.
Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan.
Entropy (Basel). 2018 Aug 16;20(8):608. doi: 10.3390/e20080608.
The main purpose of this paper is to find new estimations for the Shannon and Zipf-Mandelbrot entropies. We apply some refinements of the Jensen inequality to obtain different bounds for these entropies. Initially, we use a precise convex function in the refinement of the Jensen inequality and then tamper the weight and domain of the function to obtain general bounds for the Shannon entropy (). As particular cases of these general bounds, we derive some bounds for the Shannon entropy () which are, in fact, the applications of some other well-known refinements of the Jensen inequality. Finally, we derive different estimations for the Zipf-Mandelbrot entropy () by using the new bounds of the Shannon entropy for the Zipf-Mandelbrot law (). We also discuss particular cases and the bounds related to two different parametrics of the Zipf-Mandelbrot entropy. At the end of the paper we give some applications in linguistics.
本文的主要目的是找到香农熵和齐普夫 - 曼德布罗特熵的新估计值。我们应用詹森不等式的一些改进来获得这些熵的不同界。最初，我们在詹森不等式的改进中使用一个精确的凸函数，然后调整该函数的权重和定义域以获得香农熵（）的一般界。作为这些一般界的特殊情况，我们推导出香农熵（）的一些界，实际上这些界是詹森不等式其他一些著名改进的应用。最后，我们通过使用齐普夫 - 曼德布罗特定律（）的香农熵新界来推导齐普夫 - 曼德布罗特熵（）的不同估计值。我们还讨论了特殊情况以及与齐普夫 - 曼德布罗特熵的两个不同参数相关的界。在本文结尾，我们给出了一些在语言学中的应用。
