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Can one construct a harmonic function $f$ defined in unit disk with condition $f(0)\geq1$ such that area of $\{z\in\mathbb{D}: f(z)>0\}$ is small enough?

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  • $\begingroup$ $f$ is real-valued? The question is $\forall\epsilon>0$ $\exists f$ harmonic s.t. area$(\cdots)<\epsilon$? $\endgroup$ – Martín-Blas Pérez Pinilla Oct 20 '14 at 6:57
  • $\begingroup$ Small enough for what? $\endgroup$ – mrf Oct 20 '14 at 7:10
  • $\begingroup$ meaning for $\epsilon>0$ we want to find harmonic function $f$ such that $f(0)\geq 1$ and $area(\{z\in\mathbb{D}: \,f(z)>0\})\leq \epsilon$ $\endgroup$ – BigM Oct 20 '14 at 14:54
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This question was asked and answered at MathOverflow. The point of this answer is to provide the link and remove this from the list of unanswered questions. (I made this answer community wiki so that I don't gain reputation when I have not earned it and anyone can freely add details.)

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