Further generalizations of Hermite–Hadamard–Mercer and trapezoidal-type inequalities associated with the k-Beta function and numerical illustrations

In this paper, we introduce new generalized Hermite–Hadamard–Mercer and trapezoidal-type inequalities involving the k -Beta function. Our approach is based on Pavić’s generalized Jensen–Mercer inequality and highlights a unified way to obtain various integral estimates. The main novelty lies in extending several known results. In particular, we complement and generalize already established results in the literature by incorporating the k -Beta function and a broader Mercer-type method. With a new lemma, we also give explicit trapezoid-type error estimates for differentiable functions whose first derivatives satisfy certain convexity conditions. To illustrate the applications, we present several inequalities for special means of positive real numbers, such as power, harmonic-type, and exponential means. We also provide numerical examples and graphs to support our theoretical findings and demonstrate the accuracy of the new bounds.