On The Bi-analytic Functions And Some Problems Of Analytic Functions

Complex variable function is one of the important branches in the field of mathematicsand is widely used in elasticity mechanics, theoretical physics and celestial mechanics and soon since the beginning of20thcentury. Its main object of study lies in analytic function whichhas 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 theanalytic function, this paper attempted to expand the basic property and theorem of theanalytic function to the bi-analytic function, and then got the relationship between thebi-analytic function and the bi-harmonic function.As to the bounded analytic function, this paper, using the maximum modulus principleand the coefficient inequation of bounded analytic function, discussed the problems inestimating derivation, obtained the more advanced3rdand4thderivation estimator than theprevious ones, and expanded them to the nthderivation estimator. Thus, this paper generalizedthe estimation problem of bounded analytic function’s derivation, and then got a muchthorough expression.Last but not the least, this paper introduced the Poincaré measurement in unit disc andprobed into the creation and development of Schwarz-Pick inequation. In the meantime, thispaper analyzed the Schwarz-Pick inequation under the hyperbolic metric, got a deduction ofSchwarz-Pick inequation under the Poincaré measurement, and in this sense enriched thecontents of Schwarz-Pick.