In this research, we work on the generalization of the classical Hermite-Hadamard-Mercer(HHM) type inequalities by establishing a more generalized form with the addition of the beta function. The inequalities are very important in mathematical analysis, especially when estimating the bounds of the convex function within an interval. We also build new trapezoidal type inequalities for differentiable convex functions in terms of the beta function with more precise estimations. This research contributes important results of novel variants of fractal Holder's, Power mean and Young's inequalities. One of the most important role of these estimates is their flexibility since they are readily translatable into familiar integral estimates, like traditional integral inequalities, Riemann-Liouville(RL) fractional integral inequalities and k-Riemann-Liouville(k-RL) fractional integral inequalities. This identification places them at center stage in classical and fractional calculus. Lastly, to illustrate the applicability of our findings, we provide several applications of these newly derived inequalities, indicating their potential contribution in mathematical analysis and other related areas.
Mehtab et al. (2026) studied this question.