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On the extremal curvature and torsion of stereographically projected analytic curves. (English) Zbl 0886.30011

Let \(S\) denote the class of analytic and univalent functions \(f(z)\) defined on the unit disk \(U=\{z: |z|<1\}\) and such that \(f(0)=0\), \(f'(0)=1\). Let \(f\in S\) and let \(\Pi\) denote the stereographic projection of the image plane of \(f\) onto the unit sphere \(S^2\). For each \(r\), \(0<r<1\), let \(C_r= \{z:|z|=r\}\), \(C_r' =f(C_r)\) and \(C_r^{''} =\Pi (C_r')\). For each fixed \(\theta\), \(-\Pi< \theta\leq +\Pi\), let \(L_\theta =\{z:\arg z= \theta\}\), \(L_\theta' = f(L_\theta)\) and \(L_\theta^{''} =\Pi(L_\theta')\). The main aim of the author is to maximize and minimize the local curvature and torsion at a specified point on the stereographically projected curves \(C_r{''}\) and \(L_\theta{''}\) on the unit sphere \(S^2\). The author presents an interesting discussion connected with the asymptotic properties of the curvature and torsion functions for small values of \(r\).

MSC:

30C55 General theory of univalent and multivalent functions of one complex variable
30C75 Extremal problems for conformal and quasiconformal mappings, other methods
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