- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:47
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
29
2013
1
Monotonicity and comparison results for conformal invariants
Albert
Baernstein II
Washington University, ST. LOUIS, UNITED STATES
Alexander
Solynin
Texas Tech University, LUBBOCK, UNITED STATES
Comparison theorem, hyperbolic metric, harmonic measure, capacity
Let $a_1,\dots,a_N$ be points on the unit circle $\mathbb{T}$ with $a_j=e^{i\theta_j}$, where $0=\theta_1\le\theta_2\le\dots\le \theta_N=2\pi$. Let $\Omega=\overline{\mathbb{C}}\setminus\{a_1,\dots,a_N\}$ and let $\Omega^*$ be $\overline{\mathbb{C}}$ with the $n$-th roots of unity removed. The maximal gap $\delta(\Omega)$ of $\Omega$ is defined by $\delta(\Omega)=\max\{\theta_{j+1}-\theta_j:\,0\le j\le N-1\}$. How should one choose $a_1,\dots,a_N$ subject to the condition $\delta(\Omega)\le 2\pi/n$ so that the Poincaré metric $\lambda_\Omega(0)$ of $\Omega$ at the origin is as small as possible? In this paper we answer this question by showing that $\lambda_\Omega(0)$ is minimal when $\Omega=\Omega^*$. Several similar problems on the extremal values of the harmonic measures and capacities are also discussed.
Functions of a complex variable
Potential theory
General
91
113
10.4171/RMI/714
http://www.ems-ph.org/doi/10.4171/RMI/714