具有缺项系数的几类解析函数族的性质

The Properties of Several Classes of Analytic Functions with Missing Coefficients

S~*表示所有在单位圆盘D内解析且满足条件f(0)=f′(0)-1=0的星形函数族,K表示所有在D内解析且满足条件f(0)=f′(0)-1=0的凸函数族,P表示所有在D内解析且满足条件p(0)=1,Rep(z)>0的函数族.设P_n={p(z):p(z)= 1+a_nz^n+a_(n+1)z^(n+1)+…∈P},S_n~*-{f(z):f(z)=z+a_nz^n+a_(n+1)z^(n+1)+…∈S~*},K_n={f(z):f(z)=z+a_nz^n+a_(n+1)z^(n+1)+…∈K}.L_(S_n~*)={g(z)=ln(f(z))/z,f∈S_n~*},其中对数函数取使得ln 1=0的那个单值解析分支.该文研究了函数族S_n~*,K_n和L_(S_n~*)的性质,找出了解析函数族L_(S_n~*)的极值点与支撑点,并对S_n~*与K_n的极值点和支撑点作了一些探讨.

Let A be the class of functions, which are analytic in the unit disc D = {z ： ｜z｜ 〈 1}, with with f（0） = f＇（0） - 1 = 0. Let S＊ be the set of starlike functions, S＊ = {f（z） ∈A, Rezf＇（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 Pn = {p（z）：p（z） = 1 ＋anz^n ＋an＋1z^n＋1 ＋…∈P}, Sn＊ = {f（z） ： f（z） = z ＋anz^n ＋an＋1z^n＋1 ＋…∈ S＊}, Kn={f（z） ： f（z） = z＋anz^n＋an＋1z^n＋1＋…∈ K}. Ls＊ = {g（z） = ln f（z）/z , f∈ Sn＊}, where the logarithmic function satisfy in 1 =0. In this article the author investigated the properties of the classes Sn＊, Kn and Lsn＊. The author obstained the extreme points and support points of Ls＊, and discussed the extreme points and support points of Sn＊ and Kn.