The Verification of an Inequality
نویسنده
چکیده
In a recent paper, we verifed a conjecture of Mej́ıa and Pommerenke that the extremal value for the Schwarzian derivative of a hyperbolically convex function is realized by a symmetric hyperbolic “strip” mapping. There were three major steps in the verification: first, a variational argument was given to reduce the problem to hyperbolic polygons bounded by at most two hyperbolic geodesics; second, a reduction was made to hyperbolic polygons bounded by exactly two symmetric hyperbolic geodesics; third, for hyperbolic polygons bounded by exactly two symmetric hyperbolic geodesics a computation was made, using properties of special functions, to find the maximal value of the Schwarzian derviative. In between the second and third steps, an assertion was made that “using an extensive computational argument which considers several cases” the problem of computing the Schwarzian derivative for hyperbolic polygons bounded by exactly two symmetric hyperbolic geodesics could be reduced to computing the Schwarzian derivative for hyperbolic polygons bounded by exactly two symmetric hyperbolic geodesics under the assumption that the argument z of the Schwarzian derviative satisfied the restriction 0 ≤ z < 1. In this paper, we provide a verification for that assertion.
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