面积-容量-模不等式与填充Julia集的面积 献给杨乐教授80华诞  

Area-capacity-modulus inequality and area of the filled Julia set

作　　者：曹杰[1] 王晓光[1] 尹永成[1] Jie Cao;Xiaoguang Wang;Yongcheng Yin

机构地区：[1]浙江大学数学科学学院

出　　处：《中国科学：数学》2019年第10期1359-1372,共14页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11622108和11771387)资助项目

摘　　要：本文证明一般的面积-模不等式area(K)≤area(D)e^-4π mod(A),其中K是平面上满的正规紧集,D是包含K的有界单连通区域,区域A=D\K.上式等号成立当且仅当A是同心圆环区域.进一步,本文建立平面上满的正规紧集K的面积-容量-模不等式area(K)≤π cap(K)^2e2L-4π mod(Λ),其中L是关于K的Green函数的最大临界值,Λ是K诱导的树(集合论中的树).上式等号成立当且仅当K是闭圆盘或mod(Λ)=∞.利用此不等式,本文得到d次首一复多项式f填充Julia集K(f)面积的最佳上界估计area(K(f))≤πe^-2qL/d-q-1,其中L=max{Gf(c)|f′(c)=0},q为f在集合G^-1f(L)\K(f)上的临界指数.In this paper,we prove the general area-modulus inequality area(K)6 area(D)e^-4πmod(A),where K is a full regular compact set in the plane,D is a bounded simply connected domain containing K and A=D\K.Here the equality holds if and only if A is a round annulus.Then we establish the area-capacity-modulus inequality area(K)≤πcap(K)^2e2L-4πmod(Λ)for full regular compact set K in the plane,where L is the maximal critical value of the Green function induced by K andΛis the tree(in set theory)induced by K.Here the equality holds if and only if K is a closed disk or mod(Λ)=∞.Using this inequality,we obtain an upper bound for the area of the filled Julia set K(f)for the monic complex polynomial f of degree d.That is,area(K(f))≤πe^-2qL/d-q-1,where L=max{Gf(c)|f′(c)=0}and q is the critical index for f on the set G^-1f(L)\K(f).

关 键 词：调和测度 DIRICHLET积分 模 容量 填充Julia集 

分 类 号：G63[文化科学—教育学]