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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 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.

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    $\begingroup$ (Cross posted from MSE) $\endgroup$
    – Silvia Ghinassi
    Commented Feb 19, 2016 at 13:00

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In page 86 of "Harmonic mappings in the plane" 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.

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  • $\begingroup$ Silvia Ghinassi: Terry Sheil-Small is one person, and Sheil-Small is his last name. $\endgroup$
    – Alexandre Eremenko
    Commented Feb 19, 2016 at 13:55
  • $\begingroup$ @AlexandreEremenko thanks. And my last name is Ghinassi :) $\endgroup$
    – Silvia Ghinassi
    Commented Feb 19, 2016 at 13:57

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