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Abstract

Assume that $u$, $v$ are conjugate harmonic functions on the unit disc of $\mathbb{C}$, normalized so that $u(0)=v(0)=0$. Let $u^*$, $|v|^*$ stand for the one- and two-sided Brownian maxima of $u$ and $v$, respectively. The paper contains the proof of the sharp weak-type estimate $$ \mathbb{P}(|v|^*\geq 1)\leq \frac{1+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\ldots}{1-\frac{1}{3^2}+\frac{1}{5^2}-\frac{1}{7^2}+\ldots} \,\mathbb E u^*.$$ Actually, this estimate is shown to be true in the more general setting of differentially subordinate harmonic functions defined on Euclidean domains. The proof exploits a novel estimate for orthogonal martingales satisfying differential subordination.