Research On Related Problems Of The Bi-analytic Functions And Analytic Functions

Complex variable function is one of the important branches in the field of mathematics and is widely used in elasticity mechanics, theoretical physics and celestial mechanics and so on since the beginning of 20 th century. Its main object of study lies in analytic function which has a wonderful property and functions as an important application in many subjects.Relying on the creation background and current situation of some theories relating to the analytic function, this paper attempted to expand the basic property and theorem of the analytic function to the bi-analytic function. A kind of special bianalytic functions like A z +Bz+C(A、B and C are constant) is studied in this paper, and received several related theorems.As to the bounded analytic function, this paper, using the nature and derivative estimators bounded analytic functions, obtained 3rd and 4th derivative estimators zero function on a bounded, and using the maximum modulus principle and the coefficient inequation of bounded non-zero function and bounded analytic function, expanded them to the nth derivation estimator. And compare the results before document [1]. This paper using the nature of the relationship between positive real part coefficient function and positive real part functions to obtain nth derivative estimation formula of different functions, thus enriching study results positive real part functions of the derivative estimation.Last but not the least, this paper probed into the creation and development of Schwarz-Pick inequation. In the meantime, this paper analyzed the Schwarz-Pick inequation under the hyperbolic metric and hyperbolic metric derivative, got a hyperbolicmetricderivativeenhancedschwarz-pickinequality,andinthissenseenrichedthecontentsofschwarz-pick.