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On the Faber polynomials of the univalent functions of class \(\Sigma\). (English) Zbl 0752.30002

The \(n\)-th Faber polynomial \(\varphi_ n(t)\) of a function \(F(z)=1/f(1/z)\), \(f\in S\), is defined by the expansion \[ \log{F(z)-t\over z}=-\sum^ \infty_{n=1}{1\over n}\varphi_ n(t)z^{-n}. \] The author gives an explicit expression for \(\varphi_ n(t)\) in powers of \(t- \alpha_ 0=\varphi_ 1(t)\), and derives an inequality (*) \(|\varphi_ n'(t)|\leq n\) \(u_{2n}\), where \(u_{2n}\) is the \(2n\)-th Fibonacci number. Equality in (*) at \(t=\varepsilon\), \(|\varepsilon|=1\), only occurs if \(F(z)=z- 2\varepsilon+{\varepsilon^ 2\over z}\), that is if \(f(z)=z/(1- \varepsilon z)^ 2\) is a rotation of the Koebe function.

MSC:

30C35 General theory of conformal mappings
30C55 General theory of univalent and multivalent functions of one complex variable
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