Landau-Bloch type theorems of certain subclasses of biharmonic mappings

In this paper, we first establish a Landau-Bloch type theorem for certain bounded and normalized biharmonic mappings F(z) = |z|21(z) + h(z), where 1(z) and h(z) are harmonic in the unit disk with |1(z)| ? M1, |h(z)| ? M2. In particular, our result is sharp when M1 = M2 = 1. Then, we establish several new versions of Landau-Bloch type theorems for certain normalized biharmonic mappings with the coefficients condition in place of |h(z)| ? M2 or |1(z)| ? M1, and obtain several sharp results.