On the equivalence of a Hessian-free inequality and Lipschitz continuous Hessian

It is known that if a twice differentiable function has a Lipschitz continuous Hessian, then its gradients satisfy a Jensen-type inequality. In particular, this inequality is Hessian-free in the sense that the Hessian does not actually appear in the inequality. In this paper, we show that the converse holds in a generalized setting: if a continuos function from a Hilbert space to a reflexive Banach space satisfies such an inequality, then it is Fr\'echet differentiable and its derivative is Lipschitz continuous. Our proof relies on the Baillon-Haddad theorem.