The Singular Directions And Shared-values Of Meromorphic Functions Of Infinite Order

The thesis consists of three chapters.In Chapter 1,we mainly introduce the status of recent researches and the development of singular direction and shared-values of meromorphic function at home and oversea, 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 , 3 and 4, we mainly discuss some characters about shared-values and a singular direction of meromorphic functions of infinite orders, and we obtain three results:Theorem 1 Let f(z) be a meromorphic function of infinite order and of finite hyperorder .Then there exists a directionΔ(θ) = { z arg z=θ},0≤θ≤2Ï€such that for every small positive numberε( <Ï€/2), f ( z )and f′′( z )( k≥3) share at most two distinct finite values in the singular domain { z argz -θ<ε}.Theorem 2 Let f(z) be a meromorphic function of infinite order and of finite hyperorder.Then there exists a directionΔ(θ) = { z arg z=θ}, 0≤θ≤2Ï€such that for every small positive numberε( <Ï€/2). f ( z ) and ( ) ( )f kz ( k≥3)share at most two distinct finite values in the singular domain { z argz -θ<ε}. Theorem 3 Let f(z) be a meromorphic function of infinite order and of finite hyperorder.L(f)=b_kf^k+b_k-3f^k-3+b_k-4f^k-4+…+b₀f,and k≥3 b_j ( j = 0,1, , k - 3,k)b_k≠0,then there exists a directionΔ(θ) = { z arg z=θ},0≤θ≤2Ï€.such that for every small positive numberε( <Ï€/2). f ( z ) and L ( f ) share at most two distinct finite values in the singular domain { z argz -θ<ε}.