The Clunie Sheil Small conjecture for the second co-efficient of a univalent harmonic function on the unit disk is as follows:-
Suppose, $h(z)+\overline{g(z)}$ is a one-one harmonic function on the unit disk where $h(z)$ and $g(z)$ are analytic. Assume that $h(0)=g(0)=g'(0)=0$ and $h'(0)=1$. Then the conjecture is:- $|h''(0)|\le 5$.
This is a open problem as far as I know, but I think I heard somewhere that some bound for $|h''(0)|$ has been proved. I could find some references online where they proved a bound for this for certain class of univalent harmonic functions. But I can't find any reference where they prove some bound for $|h''(0)|$ in the general case.
Can anyone please give a reference where I can find this.