Schwarz引理的2个重要推广  

Two Important Generalizations of Schwarz Lemma

作　　者：邓键[1] 胡文正 DENG Jian;HU Wenzheng(School of Mathematical Sciences,South China Normal University,Guangzhou 510631,China)

机构地区：[1]华南师范大学数学科学学院,广州510631

出　　处：《华南师范大学学报（自然科学版）》2023年第6期109-115,共7页Journal of South China Normal University(Natural Science Edition)

基　　金：广东省自然科学基金项目(2021A1515010374)。

摘　　要：首先,利用Schwarz引理给出了Carathéodory不等式的一种证明方法;然后,对于一类在>>上全纯且以∞为本性奇点的复变函数f,得到了lim_(r→∞)A(r)/r^(n)与∞之间的关系;其次,对于满足某个条件的函数类f_(a),利用Schwarz引理得到其单叶圆盘半径,并将其推广到一类满足条件f(0)=0,f′(0)=a>0且将B(0,r)映射到自身的函数类f_(a)^(r),得到了此函数类中函数的单叶圆盘半径。A proof of Carathéodory inequality by Schwarz lemma is given,then the relation between lim_(r→∞)A(r)/r^(n) and ∞ is obtained for a class of complex functions f which are analytic on>>and take ∞ as the natural singular point.Se-condly,the radius of the univalent disk of a class of analytic functions f_(a) which satisfy some certain conditions was obtained by using Schwarz lemma.Finally,this result is extended to a class of analytic functions f_(a)^(r) which satisfy f(0)=0,f′(0)=a>0 and maps B(0,r)to itself and obtain the radius of univalent disk of functions belong to f_(a)^(r).

关 键 词：SCHWARZ引理 最大模原理 幅角原理 单叶函数 简单Jordan闭曲线 

分 类 号：O174.5[理学—数学]