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The Poincaré metric and isoperimetric inequalities for hyperbolic polygons. (English) Zbl 1075.30016

This paper proves several isoperimetric inequalities for the conformal radius (and its reciprocal, the Poincaré density). The authors obtain bounds for the conformal radius, \(R\), using the Euler gamma function, \(\Gamma\), of which the principal is the following, \[ \frac{R^2(D_n, z_0)}{1 - | z_0| ^2} \leq \frac{\Gamma^2(1 - \frac1n) \, \Gamma(\frac12 + \frac1n + \beta) \, \Gamma(\frac12 + \frac1n - \beta)}{\Gamma^2(1 + \frac1n) \, \Gamma(\frac12 - \frac1n + \beta) \, \Gamma(\frac12 - \frac1n - \beta)} . \] Here \(D_n\) is a hyperbolic polygon with \(n\) sides, while \(\beta = \frac12 - \frac1n - \frac{2A}{\pi n}\). It is also shown that equality holds above only for regular hyperbolic \(n\)-gons centered at the point \(z_0\).
This bound and several others follow from a triangulation argument introduced earlier by the third author [St. Petersbg. Math. J. 11, No. 1, 1–65 (2000; Zbl 0935.30019)]. A lower bound similar to the upper bound stated is proved using a geometric transformation called dissymmetrization, first used by V. N. Dubinin [Math. USSR, Sb. 52, 267–273 (1985; Zbl 0571.30025)]. The paper also proves certain monotonicity and convexity properties of combinations of gamma functions of the same form as the right hand side above, which may have independent interest.

MSC:

30C75 Extremal problems for conformal and quasiconformal mappings, other methods
33B15 Gamma, beta and polygamma functions
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