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 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.



Coefficient problem for univalent harmonic functions on unit disk
In page 86 of "[Harmonic mappings in the plane"](https://books.google.com/books?id=EZLQw7bPX5kC&pg=PA87&lpg=PA87&dq=clunie+sheil+small+conjecture&source=bl&ots=tggTSwxMws&sig=r2vv8DXZQKKgQ3HDLD8R83aW3Fw&hl=en&sa=X&ved=0ahUKEwjUnbyC74PLAhWDHx4KHbZ8C78Q6AEIIzAG) you can find references for these bounds.
In section 6.3 they prove the bound $|a\_2| \leq 49$. They also reference to an earlier bound proved by Clunie and Sheil-Small.


The conjecture is related to the now proved Bieberbach conjecture, so I wouldn't be surprised if more progress has been done in recent years using SLE.


