Stanisława KANAS and Toshiyuki SUGAWA

*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.

submission: 27 December, 2004

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