Can a real harmonic function on the unit disk satisfy $f(0)=1$ while the area of $\{z:f(z)>0\}$ is zero?

Question

Does there exist a harmonic function defined in the unit disk such that

(1) $f(0)=1$

(2) area of $\{z\in\mathbb{D}:\,f(z)>0\}$ is equal to zero?

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I tried to use certain representations of harmonic functions, and to relate it to analytic content.

Answer 1

Since $f$ is continuous and $f(0)=1$ then there exists $r>0$ such that the inequality $x^2 +y^2<r$ implies that $f(x,y)>\frac{1}{2} .$ Hence $$m(\{z\in\mathbb{D}: f(z)>0\} )>\pi r^2 >0,$$ where $m$ denotes the Lebesgue measure.