We use a Rényi entropy method to prove a strong converse theorem for the task of quantum state redistribution. More precisely, we establish the strong converse property for the boundary of the entire achievable rate region in the (e, q)-plane, where the entanglement cost e and quantum communication cost q are the operational rates describing a state redistribution protocol. The strong converse property is deduced from explicit bounds on the fidelity of the protocol in terms of a Rényi generalization of the optimal rates. Hence, we identify candidates for the strong converse exponents for entanglement cost e and quantum communication cost q, respectively. To prove our results, we establish various new entropic inequalities, which might be of independent interest. These involve conditional entropies and mutual information derived from the sandwiched Rényi divergence. In particular, we obtain novel bounds relating these quantities to the fidelity of two quantum states.