Extensions of Weighted Integral Inequalities for GA-Convex Functions in Connection with Fejér’s Result

This study introduces and analyzes several new functionals defined on the interval [0,1], which are associated with weighted integral inequalities for geometrically–arithmetically (GA) convex functions. Building upon the classical Hermite–Hadamard and Fejér inequalities, we define mappings such as G(u), Hyu, Kyu, Nu, L(u), Ly(u), and Syu, which incorporate a GA-convex function x and a non-negative, integrable weight function y that is symmetric about the geometric mean s1s2. Under these conditions, we establish novel Fejér-type inequalities that connect these functionals. Furthermore, we investigate essential properties of these mappings, including their GA-convexity, monotonicity, and symmetry. The validity of our main results is demonstrated through detailed examples. The findings presented herein provide significant refinements and weighted generalizations of known results in the literature.