The Singular Directions Of Meromorphic Functions And Algebroid Functions

The thesis consists of four chapters.In Chapter 1, we mainly introduce the status of recent researches and the development of the Borel directions and T directions for meromorphic functions and algebriod functions at home and overseas as well as some basic definitions and fundamental results for the value distribution theory over the complex field. Moreover we simply introduce researches and innovations of this thesis.In Chapter 2, we mainly discuss some characters for meromorphic functions with finite logarithmic order and its derivative, then we obtain two results. The one is similar to the Hayman direction:Theorem 1 Let f (z) be a transcendental meromorphic function with finite logarithmic orderλin the complex field C and there exists three finite complex numbers a , b,c which satisfy b≠0, c≠0,b≠c, then we have for each positive integer k .The other is similar to the full discs for meromorphic functions with finite order:Theorem 2 Let f (z) be a meromorphic function with finite logarithmic orderλ( 2<λ<+∞) in the complex field C,if the radial: is a Borel direction of logarithmic orderλ?1 for f (z),then there exists a list of discs: Such that f (z) takes arbitrary complex number a for at least j(log |z j|)λ?δtimes in everyΓj, at most except for some complex numbers in the two discs: S 1 , S2whose radius are e ? ej and lj i→m∞δj=1. In Chapter 3, based on some results about T direction of Zheng Jiahua [1 6] and Guo Hui [1 7],we discuss the problem of T direction for meromorphic functions with infinite order and obtain two results such as:Theorem 3 Let f (z) be a transcendental meromorphic function with orderλ=∞in the complex field C, then f (z) must has a T direction.Theorem 4 Let f (z) be a transcendental meromorphic function with orderλ=∞in the complex field C, then f (z) and its derivatives f ( k )(z), ( k =1,2,3,) have the common T directions.In Chapter 4, we obtain some results about T direction and Borel direction for algebroid functions with finite order: Theorem 5 Let w( z) be aν?valued algebroid function with infinite order, and u (r) a type function of w( z). Then for arbitrary positive numberε, (0<ε<π2),there exists a radial: B: arg z =θ0, ( 0≤θ0 <2π) such that for each complex number a in the angular regionΔ(θ0 ,ε), possibly except for at most 2νcomplex numbers.In addition, we obtain three conclusions which tie T directions to the maximality Borel directions, Borel directions of largest type as well as Borel directions.