The Singular Directions And Filling Up Discs Of The Iterated Functions

In this thesis, we investigated the Borel direction and Julia direction of the meromorphic functions and entire functions with the iterated order; we also investigated the singular direction of functions in the unit circle. It includes following five chapters.In chapter 1, we introduced the definitions and knowledge concerning iterated order of meromorphic function and entire function, such as the iterated order, the iterated exponent of convergence of zeros and the singular directions of f .In chapter 2, we investigated the argument distribution of the multiple value and the derived functions of integral functions with iterated order, and gain similar result.In chapter 3, by using the concept of iterated order, we investigated the Borel direction of the derived functions of meromorphic functions with iterated order and manifest that the direction is the function's Borel direction.In chapter 4, we investigated the filling discs of meromorphic function with iterated order in the unit circle, at the same time we use these filling discs to gain the theorems about Borel directions and Julia directions of this meromorphic function.In the last chapter, we got a result about filling up disc and the Borel directions of the K-quasiconformal meromorphic functions with iterated order by using the fundamental inequality of K-quasiconformal meromorphic functions in the plane.