We argue that a fundamental (conjectured) property of memoryless quantum channels, namely the strong superadditivity, is intimately related to the decreasing property of the quantum relative entropy. Using the latter we first give, for a wide class of input states, an estimation of the output entropy for phase damping channels and some Weyl quantum channels. Then we prove, without any input restriction, the strong superadditivity for several quantum channels, including depolarizing quantum channels, quantum-classical channels and quantum erasure channels.