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On conformal representations of the interior of an ellipse. (English) Zbl 1098.30011
The authors considered the conformal mappings \(f\) and \(g\) of the unit disk onto the inside of an ellipse with foci at \(\underline{+} 1\) so that \(f(0)=0\), \(f'(0)>0\), \(g(0)=-1\), \(g'(0)>0\). Their main interest was to show positivity of the Taylor coefficients of \(f\) and \(g\) about the origin. The authors used a clever relation between \(f\) and \(g\) and the fact that \(f\) satisfies a second order linear ODE. The authors also looked at some applications to the class of \(k\)-uniformly convex functions.

30C20 Conformal mappings of special domains
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
33E05 Elliptic functions and integrals
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