解析函数族的卷积性质与极值问题

Convolution properties and the extreme problems of analytic function classes

S_n(a,h)表示中满足条件f(0)=0=f′(0)-1,且[aD^(n+2)f(z)+(1-a)D^(n-1)f(z)]/[aD^(n-1)+f(z)+(1-a)D″f(z)]<入(z)的全体函数组成的集合,其中0≤a≤1,D″f(z)=((k*k*…*k)*f)(z),k(z)=z/(1-z)~2,h(z)是U中具有正实部的凸单叶函数.本文证明了S_(n+1)(a,h)S_n(a,h),S_n(a,h)S_n(0,h),完全决定了EHS_n(a,(1+(1-2β)z)/(1-z)),SuppS_n(a,(1+(1-2β)z)/(1-z))(0≤β<1)及SuppS(S_n(a,h)).得到了S_n族函数的系数估计、H^p性质及卷积性质.同时还引进了与S_n(a,h)类似的函数族C_n(a,h),并进行了研究。

Let S_n(a,h) denote the class of analytic functions fon the unit disk U with f (0)=0=f'(0)-1 satisfying (aD^(n-2)f(z)+(1-a)D^(n-1)f(z))/(aD^(n-1)f(z)+(1-a)D'f(z))<h(z),where 0≤α ≤1, D^n f(z)=(k*k*…*K)*f(z),k(z)=z/(1—z)~2, and h(z) is a convex univalent function on U with Reh (z)>0, It is proved that S_(n-1)(a,h)S_n(a,h),_n,(a,h)S_n(0, h). and identified EH S_n(a, (l+(2β)z)/(l-z)), Supp S_n[a,[l+(1-2β)z]/(1—z)](0≤β<l) and Supp S(S_n(a,h)). The coefficients estimation, the properties of H^p and Hadamard products for S_n(a,h) are obtained. One more such class C_n(a,h) is introduced and studied here.