The variation of the “equivocation” and the average mutual information over cascaded channels is studied, using the generalized information measures (entropies of degree α) and Renyi information measures (entropies of order α). The “equivocation inequality”, which indicates that the equivocation can never decrease as we go further from the input on a sequence of cascaded channels, is shown to be satisfied by Renyi entropies (for 0<α<1) for all channels and all probability distributions. The generalized entropies are shown to satisfy this inequality for binary symmetric channels for certain probability distributions The relations among the mutual information measures between the different terminals of the sequence of cascaded channels are studied, considering both the generalized and Renyi entropies. A necessary and sufficient condition for transmitting information without any loss across cascaded channels is obtained, when Renyi entropies are applied. The same condition is proved to be sufficient for achieving the “equivocation equality” (which indicates the case of no information loss across cascaded channels) over the class of binary symmetric channels, when the generalized entropies are applied.