## 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

### Keywords:

Faber polynomial; Fibonacci number; Koebe function; class $$S$$
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### References:

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