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?

  • $\begingroup$ $f$ is real-valued? The question is $\forall\epsilon>0$ $\exists f$ harmonic s.t. area$(\cdots)<\epsilon$? $\endgroup$ Oct 20, 2014 at 6:57
  • $\begingroup$ Small enough for what? $\endgroup$
    – mrf
    Oct 20, 2014 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, 2014 at 14:54

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