  In this paper, we obtain a sharp distortion theorem for a class of functions
in $\alpha$-Bloch spaces, and as an application of it, we establish the
corresponding Landau's theorem. These results generalize the corresponding
results of Bonk, Minda and Yanagihara, and Liu, respectively. We also prove the
existence of Landau-Bloch constant for a class of functions in Hardy spaces and
the obtained result is a generalization of the corresponding result of Chen and
Gauthier.
