吴东东, 陈行堤
(华侨大学 数学科学学院, 福建 泉州 362021)
右半平面调和映照的卷积
吴东东, 陈行堤
(华侨大学 数学科学学院, 福建 泉州 362021)
调和映照; 卷积; 凸映照; 单叶性判别; 几何特征;Cohn法则
两个调和映照的卷积映照的单叶性判别得到不少学者的研究[3-9].Dorff[5]证明了如下定理.
Dorff等[6]证明了如下的引理以及两个定理.
式(3)中:-A,-B,-C是三次多项式
的3个根(A,B,C可能相等).
下面引用Cohn法则[10]的引理B.
证明 假设-A,-B,-C是式(4)的3个根,则有
从而有|ABC|=|a/2|≤1/2<1,即-A,-B,-C这3个根中至少有一个根在单位圆盘D内.
当a∈(0,1]时,由t(z)可构造
所以有|-3a2/4-a/2+1|<|1-a2/4|.再应用t1(z)可构造
当a∈(0,1]时,有
(-2+2a+a2)2-(-4+a+2a2)2=-3(a+1)(a-1)(a+2)(a-2)≤0.
另外,当a=0时,第二复伸张w(z)=-z2为定理B中θ=π,n=2的特殊情形.因此,a∈[0,1]时定理1成立.
对h+g=z/(1-z)进行求导,可得
进一步对h′,g′求积分可得
而且有
通过Mathematica软件,f和f0*f分别把单位圆盘映成的图像,如图1所示.
(a) f (b) f0*f 图1 单位圆盘映成的图像Fig.1 Mapping unit disk onto domains
因此,可得
类似的有
所以有
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(责任编辑: 钱筠 英文审校: 黄心中)
Convolution of Harmonic Mapping in Right-Half Plane
WU Dongdong, CHEN Xingdi
(School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China)
harmonic mapping; convolution; convex mapping; univalent criterion; geometric properties; Cohn′s rule
10.11830/ISSN.1000-5013.201703026
2016-09-05
陈行堤(1976-),男,教授,博士,主要从事函数论的研究.E-mail:chxtt@hqu.edu.cn.
国家自然科学基金资助项目(11471128); 福建省自然科学基金资助项目(2014J01013); 华侨大学中青年教师科研提升计划项目(ZQN-YXl10); 华侨大学研究生科研创新能力培育计划项目(2016年度)
O174.55
A
1000-5013(2017)03-0430-05