关键词: 闭一致收敛拓扑, 极值点, 支撑点

Abstract: Let A be the class of functions, which are analytic in the unit disc D={z:|z|<1}, with f (0)=f′(0)-1=0. Let S* be the set of starlike functions, S*={f (z)∈A, Re zf′(z) / f(z)>0, z ∈ D}. Let K be the set of convex functions, K={f(z) ∈ A, Re(1+zf’’(z) / f ’(z)>0, z∈ D}. P denotes the class of functions p(z), which are analytic in D and satisfy p(0)=1, Rep(z)>0. Let P_n={p(z): p(z)=1+a_nzⁿ+a_n+1zⁿ⁺¹+… ∈ P}, S*_n={f (z): f(z)=z+a_nzⁿ+a_n+1zⁿ⁺¹+… ∈ S*}, K_n={ f(z): f (z)=z+a_nzⁿ+a_n+1zⁿ⁺¹+ … ∈K}. L_Sn*={g(z)=ln f(z) / z, f ∈ S_n*}, where the logarithmic function satisfy ln1=0. In this article the author investigated the properties of the classes S*_n, K_n and L_Sn*. The author obstained the extreme points and support points of L_Sn*, and discussed the extreme points and support points of S*_n and K_n.

Key words: Topology of uniform convergence, Extreme point, Support point