A study on the holomorphic mappings on the unit ball

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1.We give a sufficient condition that the first element of a Loewner chain on the Euclidean unit ball can be extended to a quasiconformal homeomorphism of C^n onto itself.2.We give sufficient conditions for locally biholomorphic mappings on the unit ball in a complex Banach space to be starlike of certain order or strongly starllke.3.We obtain sufficient conditions for biholomorphic mappings on the unit ball of a complex Banach space to have parametric representation.4.We consider the generalization of the Loewner differential equation to reflexive complex Banach spaces. We obtain some Lipechitz continuity results related to Loewner chains and their transition mappings.5.We give a sharp lower bound for the distortion of the image of the Roper-Suffridge extension operator : This result gives a positive answer to an open problem posed by Liczberski-Starkov.6.We give a complete classification of rational proper holomorphic maps from B^n into B^<2n> when n>3.7.Pescar investigated the univalence of certain integral operators. We show that the results are obtained by the Schwarz lemma. We also give some generalizations.8.Let f(z,t) be a subordination chain on[0,α] on the Euclidean unit ball B in C^n. We give a sufficient condition that f can be extended to a quasiconformal homeomorphism of R^<2n> onto itself.9.We give coefficient conditions for some rational mappings to be starlike or convex on the Euclidean unit ball B in C^n.