Abstract
We derive a second variation formula of the Koebe function developed by Bombieri by using a linear version of the Loewner differential equation. We then express the formula in terms of classical orthogonal polynomials to consider the Bombieri numbers obtained by Greiner and Roth (Proc Am Math Soc 129(12):3657–3664, 2001) and by Prokhorov and Vasil’ev (Georgian Math J 12(4):743–761, 2005). We show a question raised by Bombieri on these numbers has a negative answer in the low order cases.
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