The Study On Some Univalent Functions On Unit Disk

In this dissertation,the analytic characterization conditions and coefficient estimate of some subclasses of univalent functions on the unit disk D=z∈C:|z|<1} are studied.Let S(p)denote the collection of univalent meromorphic functions f(z)on the unit disk with a simple pole at z=p∈(0,1)and satisfy the normalization conditions f(0)=0,f’(0)=1.When f∈S(p),its Taylor expansion is f(z)=z+∑n=2∞anzn,|z|<p,and Laurent expansion is f(z)=∑n=-1∞bn(z-p)n,|z-p|<1-p.In the course of studying meromorphic functions S(p),Clunie,Miller et al.proposed and studied two important subclasses:concave functions and starlike meromorphic functions with respect to ω0.In this dissertation,the two subclasses are generalized,the analytic characterization conditions are given and the Taylor and Laurent coefficients are estimated.The concave function class Co(p)is the collection of functions in S(p)that map the unit disk onto the complement of bounded closed convex domain.Let α∈[0,1),we call f∈S(p)the concave function of order α,if f maps the unit disk onto the complement domain of the closed bounded convex domain of order α,and we write for Co(p,α).In this dissertation,the necessary and sufficient conditions that f∈S(p)is concave function of order α is characterized and the Taylor coefficients are estimated.When ω0(ω0≠0,∞),the class ∑*(p,ω0)is the collection of the starlike meromorphic functions with respect to ω0 in S(p)that map the unit disk onto the complement domain of the closed bounded starlike domain with respect to ω0.Let α∈[0,1),we call f∈S(p)the starlike meromophic function with respect to the point ω0 of order α,if f maps the unit disk onto the complement domain of the closed bounded starlike domain with respect ω0 of order α,and we write for ∑*(p,ω0,α).In this dissertation,the necessary and sufficient conditions that f∈S(p)is starlike meromophic function with respect toω0 of order α is characterized and the Taylor coefficients and Laurent coefficients are estimated.Analytic functions on the unit disk are also a hot research area,the class A denote the collection of analytic functions f(z)on the unit disk and satisfy the normalization conditions f(0)=0,f’(0)=1.When f∈A,its Taylor expansion is f(z)=z+∑n=2∞anzn,|z|<1.When α∈R,let M(α)denote the collection of f∈E A which satisfly the condition R((1-α)zf’(z)/f(z)+α(1+zf"(z)/f’(z)))>0,the class M(α)was proposed by Mocanu.In this dissertation,we generalize α-convex function class M(α),Let β∈[0,1),α≥0,we call f∈A the α-convex function of order β,if f satisfies the condition R((1-α)zf’(z)/f(z)+α(1+zf"(z)/f’(z)))>β,and we write for Mαβ.In this dissertation,we prove the f∈Mαβ is Bazilevic-type function and obtain the estimate of Taylor coefficients.