A New Refinement of the Jensen Inequality with Applications in Information Theory

The Jensen inequality is one of the most important inequalities in theory of inequalities, and numerous results are devoted to this inequality. This inequality has many applications in several fields. This article is devoted to present a new interesting refinement of the well-known Jensen inequality and to give applications for means. The paper also aims to achieve numerous applications in information theory; in particular, a comprehensive detail has been given for Zipf’s law and obtained bounds for Zipf–Mandelbrot entropy. At the end of the article, a more general refinement of Jensen inequality is presented associated with m finite sequences.