This paper investigates generalizations of the classical Jensen-Steffensen inequality to the operator setting of divided differences. Using Hermite–Genocchi representation and leveraging the fundamental connection between the n-convexity of a function f and the convexity of its n-th derivative \(f^{(n)}\) , we establish a series of new integral and discrete inequalities, including versions of Szegő’s and Mercer’s inequalities for divided differences. Furthermore, we establish a reverse inequality and a refinement theorem, thereby providing a comprehensive treatment of this generalization. A notable application of this framework is obtained by integrating the main inequality over the entire polytope of valid Jensen-Steffensen weights. This procedure yields a novel integral inequality of the Hermite-Hadamard type, providing a tight bound for the averaged operator in terms of the values at the endpoints of the monotonic sequence of nodes.
