Maximum of a function of one variable - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T14:49:05Z http://mathoverflow.net/feeds/question/108676 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108676/maximum-of-a-function-of-one-variable Maximum of a function of one variable Alexandre Eremenko 2012-10-03T00:28:28Z 2013-03-27T15:33:13Z <p>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.</p> <p>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 $f'(0)>0$.</p> <p>Is it true that maximum $f'(0)$ is achieved when $\theta=\pi/4$ ?</p> <p>There is a strong computer evidence for this, as well as the general considerations (where else can the maximum be?).</p> http://mathoverflow.net/questions/108676/maximum-of-a-function-of-one-variable/120652#120652 Answer by Matti Vuorinen for Maximum of a function of one variable Matti Vuorinen 2013-02-03T06:28:58Z 2013-02-03T06:28:58Z <p>Dear Alexandre,</p> <p>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</p> http://mathoverflow.net/questions/108676/maximum-of-a-function-of-one-variable/125733#125733 Answer by Alexandre Eremenko for Maximum of a function of one variable Alexandre Eremenko 2013-03-27T15:33:13Z 2013-03-27T15:33:13Z <p>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.</p>