TITLE: Bounding Taylor coefficients of $f(z)$ with $f(0)=1$, $f(z)\ne 0$ for $|z|\le 1$
QUESTION [6 upvotes]: 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?

REPLY [5 votes]: 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.