科研之友
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摘要
Let E be a continuum in the closed unit disk vertical bar z vertical bar %26lt;= 1 of the complex z-plane which divides the open disk vertical bar z vertical bar %26lt; 1 into n %26gt;= 2 pairwise nonintersecting simply connected domains D-k such that each of the domains D-k contains some point a(k) on a prescribed circle vertical bar z vertical bar = rho, 0 %26lt; rho %26lt; 1, k = 1, ... , n. It is shown that for some increasing function Psi, independent of E and the choice of the points a(k), the mean value of the harmonic measures %26lt;br%26gt;Psi(-1)k [1/n Sigma(k)(k=1) Psi(omega(a(k), E, D-k))] %26lt;br%26gt;is greater than or equal to the harmonic measure omega(rho, E*, D*), where E* = {z : z(n) is an element of [-1,0]) and D* = {z : vertical bar z vertical bar %26lt; 1, vertical bar arg z vertical bar %26lt; pi/n}. This implies, for instance, a solution to a problem of R. W. Barnard, L. Cole, and A. Yu. Solynin concerning a lower estimate of the quantity inf (E) max(k=1), ..., (n) omega(a(k), E, D-k) for arbitrary points of the circle vertical bar z vertical bar = p. These authors stated this hypothesis in the particular case when the points are equally distributed on the circle vertical bar z vertical bar = rho.
- 出版日期2012-7
