Subadditivity for Analytic Capacity Disjoint Compacts separated by a Line

Asked Apr 12 '14 at 22:18

Active Aug 09 '15 at 18:59

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The following problem is asked in Greene and Krantz, Problem 9, page 382:

Suppose that $C_1$ and $C_2$ are disjoint compact sets in $\mathbb{C}$ that can be separated by a line $l$ with $C_1 \cap l = C_2 \cap l = \emptyset$. Show that $$\gamma(C_1 \cup C_2) \leq \gamma(C_1) + \gamma(C_2).$$

Here, $\gamma(C)$ is the analytic capacity of the compact set $C\subset \mathbb{C}$.

All my ideas to solve this problem reach a dead end pretty quickly. Two ideas initially felt right. One was Schwarz reflection. The other was finding an open disk containing $C_1$ but that is disjoint with $C_2$, then use the Cauchy integral formula on this disk to help define a function that is holomorphic on $\mathbb{C} \backslash C_2$ that has norm $\leq$ 1. I don't think either of these ideas are useful.

edited Aug 02 '15 at 14:59

Daniel Fischer

206,697

asked Apr 12 '14 at 22:18

wellfedgremlin

1,153

What does it mean $\gamma(C)$? – vesszabo Apr 22 '14 at 17:14
That is the analytic capacity of C. – wellfedgremlin Apr 24 '14 at 08:56
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I would be very interested in seeing a proof of this. I have no idea how to prove something like that. My Ph.D. thesis was on analytic capacity, and I've never seen or heard of a similar result. If I had to guess, I would say that it is still open, but who knows... Maybe one should contact Greene or Krantz? – Malik Younsi Aug 05 '15 at 01:32
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@MalikYounsi I e-mailed Greene a couple of days ago, but no answer up to this moment... – João Ramos Aug 07 '15 at 14:26
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@JoãoRamos Did you get a reply eventually? – Daniel Fischer Dec 24 '15 at 16:04

Subadditivity for Analytic Capacity Disjoint Compacts separated by a Line

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