# Maximum of a function of one variable

Let $D$ be a circular quadrilateral (that is a Jordan region whose boundary consists of 4 arcs of circles all orthogonal to the unit circle) whose interior angles are all equal to 0, the vertices lie on the unit circle, and $D$ is inside the unit disc. Suppose also that $D$ is symmetric with respect to the real and imaginary axes, and has one vertex $z_1=\exp(i\theta)$ where $\theta\in (0,\pi/2)$. Then this number $\theta$ determines such a $D$ completely.

Let $f$ be the inverse of the Riemann map, so that $f$ is the conformal map from the unit disc onto $D$, $f(0)=0$ and $f'(0)>0$.

Is it true that maximum $f'(0)$ is achieved when $\theta=\pi/4$ ?

There is a strong computer evidence for this, as well as the general considerations (where else can the maximum be?).

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Alexandre, do you mean to say that $D$ is an ideal hyperbolic quadrilateral in the Poincare disk? (In other words, the arcs of $D$ are orthogonal to the unit circle). Otherwise, there is another parameter needed to define $D$, namely the angle that the arcs of $D$ make with the unit circle. –  Jeremy Kahn Oct 7 '12 at 1:13
Alexandre, have you looked at Chapter V of Nehari's book Conformal Mapping? He considers the general problem of finding the "Schwarz-Christoffel" mapping for hyperbolic polygons. It's complicated in general, but in your case, you should be able to write it down. –  Richard Kent Oct 7 '12 at 3:08
–  Richard Kent Oct 7 '12 at 3:09
Jeremy, you are right, I consider an ideal hyperbolic quasrilateral, and I made a correction in my description. –  Alexandre Eremenko Oct 7 '12 at 14:06
Richard, I know the literature, at least the classical books. This is probably the simplest problem on conformal mapping where there is no explicit solution. You can look in my preprint arXiv:1110.2696, where I wrote all I know abut this, and computed the thing numerically. –  Alexandre Eremenko Oct 7 '12 at 14:10

Dear Alexandre,

These ideal hyperbolic quadrilaterals are sometimes called Lambert quadrilaterals. See e.g. pointers to the literature in arXiv:1203.6494v2 [math.MG] 10 Apr 2012 .Matti Vuorinen

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Matti, it is interesting to know that they have a name. Do you know why Lambert? Lambert was an XVIII century mathematician who is remembered for Lambert series. Why is his name attached to these quadrilaterals? (Unless there was some other Lambert). –  Alexandre Eremenko Feb 4 '13 at 5:54

Alexandr Solynin told me that he solved this problem (even the more general one, for hyperbolic n-gons with all zero angles) in 1993. A. Solynin, Some extremal problems for circular polygons, (Russian) Zapiski Nauchnyh Seminarov POMI 206, 1993, 127-136. English translation is in Journal of Math Sci. 80 4, 1996, 1956-1961.

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