On conformal representations of the interior of an ellipse. (English) Zbl 1098.30011

Kanas, Stanisława; Sugawa, Toshiyuki

Ann. Acad. Sci. Fenn., Math. 31, No. 2, 329-348 (2006).

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.

Reviewer: Maslina Darus (Selangor)

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

Keywords:

conformal representation; Taylor coefficients; univalent functions; second order linear ODE; k-uniformly convex functions