In this paper, we first propose a Z(p)-eigenvalue of a tensor, which includes the Z(1)- and Z(2)-eigenvalue as its special case, and then present a Z(p)-eigenvalue bound. In particular, we give a Z-spectral radius bound for an irreducible nonnegative tensor via the spectral radius of a nonnegative matrix. The proposed bounds improve some existing ones. Some numerical examples are given to show the validity of the proposed bounds.