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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.

##### MSC:
 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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