Hadamard products of Schlicht functions and the Pólya-Schoenberg conjecture
Hadamard products of Schlicht functions and the Pólya-Schoenberg conjecture
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DOI：
10.1007/bf02566116
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发表时间：
1973-12
影响因子：
0.9
通讯作者：
S. Ruscheweyh;T. Sheil‐Small
中科院分区：
文献类型：
--
作者：
S. Ruscheweyh;T. Sheil‐Small
如果函数~0(Z)是单位圆盘Izl；1}到凸域上的Schlicht保形映射，则称函数~0(Z)是凸的；如果函数f(Z)是关于原点的圆盘到星形区域的Schlicht保形映射，则称f(Z)是星形的。在整篇文章中，我们将假设我们的凸函数和星形函数在z=0时为零。将两个级数f(Z)=~a，z“和g(Z)=~b，z”的Hadamard积或卷积定义为级数(f*g)(Z)=Z~a，b，z‘。1958年，G.P61ya和IJ勋伯格[2]提出了如下猜想。
A function~ 0 (z) is said to be convex if it is a schlicht conformal mapping of the unit disc Izl< 1} onto a convex domain, f (z) is said to be starlike if it is a schlicht conformal mapping of the disc onto a domain starlike with respect to the origin. Throughout the paper we shall assume that our convex and starlike functions vanish at z= 0. The Hadamardproduct or convolution of two power seriesf (z)=~ a, z" and g (z)=~ b, z" is defined as the power series (f* g)(z)= Z~ a, b, z'. In 1958 G. P61ya and IJ Schoenberg [2] made the following conjecture.