Complex multiplex networks consist of several subnetwork layers, which
interact via pairwise inter-layer connections. Relay synchronization between
distant layers which are not directly connected, but only via a relay layer,
can be observed in multiplex networks. We study three-layer networks of
discrete logistic maps, where each individual layer is a nonlocally coupled
ring, and demonstrate scenarios of relay synchronization of complex patterns in
the outer layers which interact via an intermediate layer. We find regimes of
relay synchronization for chimera states, i.e., patterns of coexisting coherent
and incoherent domains, and a transition from phase chimeras to amplitude
chimeras for increasing inter-layer coupling. We determine analytically the
approximate critical coupling strengths for the existence of phase chimeras
