An asymptotic property of univalent mappings (Q1337425)

The main result is as follows: Let \(f\) be a univalent holomorphic function in the upper halfplane. Then, for almost all real \(x\), there exists \(c= c(x,\vartheta)> 0\) and \(r> 0\) such that \[ |\text{Im } f(u+ iv)| > cv\quad\text{for}\quad | u- x|< v\tan\vartheta,\qquad 0< v< r. \] One ingredient of the proof is the Hayman-Wu theorem. The result indirectly adds support to the following conjecture: If \(h\) is a bounded real harmonic function in the upper halfplane which is not identically zero then \[ \limsup_{y\to 0} | h(x, y)|/y> 0 \] for almost all real \(x\). Note that this conjecture is about the linear (and not angular) approach.