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In this paper we deal with two quantum relative entropy preserver problems onthe cones of positive (either positive definite or positive semidefinite)operators. The first one is related to a quantum R\'enyi relative entropy likequantity which plays an important role in classical-quantum channel decoding.The second one is connected to the so-called maximal $f$-divergences introducedby D. Petz and M. B. Ruskai who considered this quantity as a generalization ofthe usual Belavkin-Staszewski relative entropy. We emphasize in advance thatall the results are obtained for finite dimensional Hilbert spaces.

Maps on positive operators preserving R\'enyi type relative entropies and maximal $f$-divergences