Some Properties On Operator-valued Functions

The theory of operator functions is a new direction in the theory of functions. This direction has been developed quite well since it was founded and it has been infiltrated in other branches of mathematics progressively. Based on the fundamental works done by professor Fan Ky, who was the founder of this direction, the present paper introduces the definitions of the class of meromorphic univalent operator-valued functions with positive operator coefficients on Hilbert space and the argument of the operator-valued functions, improves and extends some problems in classical geometry theory of complex functions to the case of operator-valued functions, and studies them respectively. There are two aspects in the work. Firstly, we introduce the class of meromorphic univalent operator-valued functions with positive operator coefficients on Hilbert space, obtain a sufficient and necessary condition and operator coefficients estimates for , and show that the class is closed under arithmetic means and convex linear combinations. Secondly, we introduce the integral operator , where is an operator-valued functions, and derive some argument properties of the integral operator and some interesting corollaries as the special case. Moreover, some open questions are pointed out.