Improvements to Slater's inequality and their applications via functions whose fourth-order derivatives are convex

Mathematical inequalities are highly valued for their significant properties and broad range of applications, offering diverse approaches to solving problems across various domains. This article aims to enhance Slater's inequality for functions with convex fourth-order derivatives by employing concepts of convexity and Jensen's inequality. As applications of these improvements, we present new inequalities for power means and provide several estimates for different divergences. These results are further validated graphically, demonstrating their importance. Additionally, the findings are applied to derive bounds for the well-known Zipf–Mandelbrot entropy.