On conformal representations of the interior of an ellipse

Abstract

We consider the conformal mappings $f$ and $g$ of the unit disk onto the inside of an ellipse with foci at $\pm1$ so that $f(0)=0, f'(0)>0, g(0)=-1$ and $g'(0)>0.$ The main purpose of this article is to show positivity of the Taylor coefficients of $f$ and $g$ about the origin. To this end, we use a special relation between $f$ and $g$ and the fact that $f$ satisfies a second-order linear ODE. Some applications are given to the class of $k$-uniformly convex functions.

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