New Results on Parameterized Mercer type Inequalities: Analysis and Applications (European Journal of Pure and Applied Mathematics, 2026)
In this paper, we establish a Mercer-parameterized integral identity for differentiable functions. Using this identity, we derive a family of new parameterized Hermite--Hadamard-Mercer type inequalities for differentiableconvex functions. A key feature of the proposed results is their flexibility, which allows them to recover, as special cases, Mercer-type inequalities of trapezoidal, midpoint, Bullen-type, and Simpson-type. The validity and sharpness of the obtained inequalities are illustrated through mathematical examples and graphical representations. Moreover, the derived results extend several known inequalities in the existing literature. Finally, applications to the modified Bessel function, the q-digamma function, special means, and numerical quadrature schemes are presented.
Keywords: Mercer-midpoint type inequalities; convex function; power-mean inequality; Mercer bullen type inequalities; Mercer Simspon's type inequalities; modified Bessel function; q-digamma function; quadrature schemes