Abstract
In 2021, Ullah et al., introduced a new approach for the derivation of results for Jensen's inequality. The purpose of this article, is to use the same technique and to derive improvements of Slater's inequality. The planned improvements are demonstrated in both discrete as well as in integral versions. The quoted results allow us to provide relationships for the power means. Moreover, with the help of established results, we present some estimates for the Csiszar and Kullback-Leibler divergences, Shannon entropy, and Bhattacharyya coefficient. In addition, we discuss some additional applications of the main results for the Zipf-Mandelbrot entropy.