New variants of the Hermite–Hadamard–Mercer type estimates involving the Beta function with numerical analysis

A refined form of the Jensen–Mercer inequality has recently been introduced in the literature. Utilizing this improved inequality, we derive several new variants of the Hermite–Hadamard–Mercer type inequality, along with related estimates involving the beta function. These results serve to extend and generalize various existing inequalities. To demonstrate the validity of our findings, we provide numerical examples supported by both graphical and computational analyses. In addition, we explore applications of our results to specific means, including the arithmetic, harmonic, geometric, logarithmic, and p-logarithmic means. We hope that this work will stimulate further investigations in this area.