If a continuous map between a Hilbert and a reflexive Banach space satisfies the Hessian-free Jensen inequality, then it is Frechet differentiable and its derivative is Lipschitz continuous.
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On the equivalence of a Hessian-free inequality and Lipschitz continuous Hessian
If a continuous map between a Hilbert and a reflexive Banach space satisfies the Hessian-free Jensen inequality, then it is Frechet differentiable and its derivative is Lipschitz continuous.