# Area growth of harmonic functions

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?

• $f$ is real-valued? The question is $\forall\epsilon>0$ $\exists f$ harmonic s.t. area$(\cdots)<\epsilon$? – Martín-Blas Pérez Pinilla Oct 20 '14 at 6:57
• Small enough for what? – mrf Oct 20 '14 at 7:10
• meaning for $\epsilon>0$ we want to find harmonic function $f$ such that $f(0)\geq 1$ and $area(\{z\in\mathbb{D}: \,f(z)>0\})\leq \epsilon$ – BigM Oct 20 '14 at 14:54