Advances in Fractional Bullen-Mercer Type Inequalities: Analysis via Differentiable $ s $-Convex Functions and Numerical Applications

In mathematical analysis, the theory of inequalities plays a crucial roledue to its wide applications across various fields of physical sciences. Inthis article, we develop a new class of fractional Bullen-type inequalitiesrelated to the Jensen-Mercer inequality. To achieve this, first, we obtain ageneral fractional Bullen-Mercer identity that plays the foundation for ourmain results. Additionally, using the fractional Bullen-Mercer equality andapplying properties of $s$-convex functions, we give several newinequalities using H"{o}lder's, power-mean and Young's inequalities. Someknown results are recaptured and several special cases are discussed indetail. Furthermore, applying Lipschitzian and bounded functions in ourgeneral fractional Bullen-Mercer identity, we present some new inequalities.Moreover, we offer some applications of our results, including their use inspecial means, error bounds, matrix inequality and the $q$-digamma function.To validate our findings, we perform various simulations.