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?).



Maximum of a function of one variable
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.


