Bloch Constants And Composition Operators On Bloch Spaces

In this thesis, we investigate Bloch constant Bn of the functions whose derivative has zeros with multiplicity at least n, and composition operators on Bloch spaces.1, Functions whose derivative has zeros with multiplicity at least n and related Bloch constants.We discuss the distribution of zeros of the derivative of Bloch functions whose derivative has zeros with multiplicity at least n, and improve the result of Ahlfors, Liu and Minda about the lower bound of Bloch constants Bn related to these functions. We obtain2, Composition operators on Bloch spacesFirst, we discuss the seminorm and norm of composition operators on Bloch space. The relations of seminorm and norm with angle derivative and hyperbolic metric in unit disk are obtained. In particular, when : D --D has angle derivative at some point, we show that the half norm of the composition operator induced by is 1.Second, we discuss composition operators on Bloch space with closed range. By using a distortion theorem of Bonk, Minda and Yanagihara about Bloch functions, we obtain the sharp estimation of the Lipschitz continuity of the dilation of Bloch functions. Then, we improve a theorem of Ghatage, Yan and Zheng about composition operators on Bloch space with closed range.At last, we give answers to two questions proposed by Madigan and Matheson. New criterion for compactness is obtained: A Composition operator C on little Bloch space BO is compact if and only if log belong to B0 uniformly for w D. We give an example of 97, such that C is compact on little Bloch space B0 and the set D has infinite many points.