Nevanlinna Direction In Which Deficient Function Can Be Rational Function And Distortion Theorem

The singular direction of the meromorphic function is one of the main reasearch contents of the angular distribution theory . Deviation theorem is a kind of basic val-uation theorem in complex analysis,which is widely used in many scientific fields. In this paper, we work for proving the existence of Nevanlinna direction in which the N-evenlinna directions are denned by rational functions as deficient function and explain the distortion theorem of some special domains under the Poincare metric.In this paper ,under the inspiration of Sun DaoChun’s redefined deficient value ,we can have the definition of deficient fuction when the deficient value is extended to the case of small function .At the same time ,the definition of the Nevanlinna direction of the deficient function to be rational function is also obtained.From these new definition we proved that at least a Nevanlinna direction exist while meromorphic function ω(z) meets condition for growth limr�'(T(r,ω))/ln2r=∞.Inspired by koebe theorem ,we study the distortion qualities from Ω1= {x + iy|x≥ —1/2,y∈R}、Ω2= {z ∈C\Rez ≥—1,—π、4≤argz≤π/4}、square domain to the simply connected domain,and some new results about the distortion are obtained...