The Optimal Operator Formulation of The Hermite-Hadamard and Fejér Inequalities

This paper establishes new operator versions of the Hermite-Hadamard and Fejér inequalities for the class of operator (h,m)-convex functions on Hilbert spaces. Operator (h,m)-convex functions generalize operator convex, operator m-convex and operator h-convex functions by including a nonnegative function h and a parameter m∈(0,1]. The results presented not only generalize and improve several known inequalities but also represent best possible generalizations for certain parameter selections such as h(μ)=μ and m=1. Additionally, Fejér type inequalities are derived by integrating symmetric weights, demanding integrability of the product of functions on operator domains. The approach generalizes integral inequality techniques in a classical manner to operators, offering new perspectives for functional analysis and operator theory.