Preservation of a quantum Renyi relative entropy implies existence of a recovery map

It is known that a necessary and sufficient condition for equality in the data processing inequality (DPI) for the quantum relative entropy is the existence of a recovery map. We show that equality in DPI for a sandwiched R\'enyi relative $\alpha$-entropy with $\alpha>1$ is also equivalent to this property. For the proof, we use an interpolating family of $L_p$-norms with respect to a state.