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On a problem of Pólya and Szegö. (English) Zbl 1010.30007

Let \(f\) be meromorphic in the unit disc \(E\), \(R_f(z)=|f'(z) |(1-|z|^2)\) its inner mapping radius at \(z\in E\) and \(M_f\) the set of critical points of \(R_f\). The author makes a local bifurcation research on the set \(M_f\), for the parametrization \(f_r(z)=f(rz)\), \(r>0\), and uses his results to give a new proof of the following known result: Let \(\{f,z\}\) denote the Schwarzian derivative of \(f\) and \(f\) fulfill Nehari’s univalence criterion \((1-|z|^2)^2|\{f,z\}|\leq 2\), \(z\in E\). Then the union of \(M_f\) and of the set of poles of \(f\) contains at most one element unless \(f(E)\) is a strip.

MSC:

30C35 General theory of conformal mappings
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