Evolutionary Approach to Inequalities of Hermite–Hadamard–Mercer Type for Generalized Wright’s Functions Associated with Computational Evaluation and Their Applications

The theory of integral inequalities has a wide range of applications in physics and numerical computation, and plays a fundamental role in mathematical analysis. The present study delves into the attractive domain of Hermite–Hadamard–Mercer (H-H-M)-type inequalities having a special emphasis on Wright’s general functions, referred to as Raina’s functions in the scientific literature. The main goal of our progressive study is to use Raina’s Fractional Integrals to derive two useful lemmas for second-differentiable functions. Using the derived lemmas, we proved a large number of fractional integral inequalities related to trapezoidal and midpoint-type inequalities where those that are twice differentiable in absolute values are convex. Some of these results also generalize findings from previous research. Next, we provide applications to error estimates for trapezoidal and midpoint quadrature formulas and to analytical evaluations involving modified Bessel functions of the first kind and q-digamma functions, and we show the validity of the proposed inequalities in numerical integration and analysis of special functions. Finally, the results are well-supported by numerous examples, including graphical representations and numerical tables, which collectively highlight their accuracy and computational significance.