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Let $f(z)=1+a_1z+\ldots+a_nz^n+\ldots$ be a complex analytic function defined on the unit disk $|z|\le 1$. Suppose $f(z)\ne 0$ for $|z|< 1$. I would like to know what kind statements one can make concerning bounding the absolute values of $a_n$ from above. As the example of $f(z)=(1+z)^n$ shows individual $a_i$ can be as large as you want, but I wonder still if something non-trivial can be said about the set of possible $a_i$. Maybe there are some classical results on this topic?

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    $\begingroup$ What immediately comes to mind is of course 'de Branges's theorem'/'Bieberbach conjecture', but since that theorem requires one more assumption (i.e. injectivity), I won't make this an answer. $\endgroup$ Feb 24, 2018 at 11:54
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    $\begingroup$ More assumptions are needed to give any bounds. This is not a normal family that you consider. $\endgroup$ Feb 24, 2018 at 14:57
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    $\begingroup$ Your assumption (that the function is defined on the unit disk) implies that the radius of convergence is at least 1, of course, so that you have the bound $a_n=O((1+\epsilon)^n)$ for any $\epsilon>0$. $\endgroup$ Feb 24, 2018 at 16:53
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    $\begingroup$ for fixed $i$, $a_i$ may take any value (consider $(1+cz)^n$ for $c\in (0,1)$ and large $n$) $\endgroup$ Feb 25, 2018 at 13:56

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The best conjectured in this generality seems to be the conjecture of Lewandowski and Szynal

Lewandowski, Zdzisław; Szynal, Jan, The Landau problem for bounded nonvanishing functions, J. Comput. Appl. Math. 105, No.1-2, 367-369 (1999). ZBL0945.30011.

They conjecture that for any non-vanishing $f$ in the unit disk such that $|f| < 1$ and any $n$ there exist an $L(n)$ such that $|a_0 + \dotsc + a_n| < L(n).$

As far as I can tell, this is completely open, but some results for small $n$ are known under the additional assumption that the Taylor coefficients are real:

Koulorizos, Philippos; Samaris, Nikolas, The Landau problem for nonvanishing functions with real coefficients, J. Comput. Appl. Math. 139, No.1, 129-139 (2002). ZBL1009.30010.

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    $\begingroup$ @PeterHeinig Oops, I missed a hypothesis, you are right. $\endgroup$
    – Igor Rivin
    Feb 25, 2018 at 13:17

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