The Optimal Operator Formulation of The Hermite-Hadamard and Fej´er Inequalities

This paper establishes new operator versions of the Hermite-Hadamard and Fej\'er 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 \in (۰,۱]. The results presented not only generalize and improve several known inequalities but also represent best possible generalizations for certain parameter selections such as h(\kappa)=\kappa and m=۱. Additionally, Fej\'er 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.

کلیدواژه ها:

(h ، m)-convexity ، Operator convexity ، Hermite-Hadamard inequality ، Fej&eacute ، r inequality ، Hilbert space