Kanas, Stanisława; Sugawa, Toshiyuki On conformal representations of the interior of an ellipse. (English) Zbl 1098.30011 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) Cited in 2 ReviewsCited in 10 Documents 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 PDF BibTeX XML Cite \textit{S. Kanas} and \textit{T. Sugawa}, Ann. Acad. Sci. Fenn., Math. 31, No. 2, 329--348 (2006; Zbl 1098.30011) Full Text: EuDML