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The Clunie Sheil Small conjecture for the second coefficient of a univalent harmonic function on the unit disk is as follows: Suppose, $h(z)+\overline{g(z)}$ is a one-to-one harmonic function on the ...

The harmonic majorization for a subharmonic function $h$ is well-known for bounded regions $\Omega \subset \mathbb{C}$: $$h \le 0 \text{ in }\partial \Omega \Longrightarrow h \le 0 \text{ in }\Omega.$$...

I've been reading up on Hardy spaces and (sub)harmonic functions over the open unit disk $\mathbb{D}\subset\mathbb{C}$, and I've found myself working with atypical objects in mostly-typical situations....