International Symposium on Information Theory and its Applications(ISITA2008), Auckland, New Zealand, 7-10, December, 2008 -- (Abstract)We introduce a generalized Wigner-Yanase skew information and then derive the trace inequality related to the uncertainty relation. This inequality is a non-trivial generalization of the uncertainty relation derived by S.Luo for the quantum uncertainty quantity excluding the classical mixture. And we introduce a generalized Fisher information and then derive a generalized Cram´er-Rao inequality. We also give an example for our generalized Fisher information and then derive the uncertainty relation for two observables.