*[This is a typo, now removed. -T]*

Here is something nice. Let . Then we have it's like we pinch off a tiny bit of the value at the origin with the derivative. And the shape is a Cardioid. This taken from a paper "Topology of Quadrature Domains" https://arxiv.org/abs/1307.0487

]]>Maybe the floor plan of a room — all floor plans of houses — are conformally equivalent. We can partition a rectangle into various rooms connected by doors. And then we randomly walk around. And these are all related.

]]>For “simple” univalent functions (eg those with real coefficients and more generally the typically real functions on the unit disk which may not be univalent – since they have a simple integral representation with respect to a positive unit measure on the circle, or for that matter close to convex and related univalent functions which also have nice integral representations) Bieberbach’s conjecture is fairly easy to deduce; also Littlewood proved fairly easily in the 20’s the inequality a(n)<en so the right order of the magnitude for the coefficients is not that deep a problem either.

However there are some functionals (related to the Milin one(s) that is used in de Branges' proof) on the S class that have different asymptotic behaviors on odd versus even coefficients (see Grinshpan' survey article in Kuhnau's handbook of Geometric Function Theory I, p 313) that hint to the reasons no one so far managed to prove Bieberbach without going through the 2-symmetrization of the S-class that leads to an odd univalent function and the Robertson conjecture which then follows from the negativity of the Milin functional

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