Koebe 1/4 theorem and inequalities inN=2 supersymmetric QCD

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Abstract
The critical curve C on which Imτ^=0, τ^=aD/a, determines hyperbolic domains whose Poincaré metric can be constructed in terms of aD and a. We describe C in a parametric form related to a Schwarzian equation and prove new relations for N=2 supersymmetric SU(2) Yang-Mills theory. In particular, using the Koebe 1/4 theorem and Schwarz's lemma, we obtain inequalities involving u, aD, and a, which seem related to the renormalization group. Furthermore, we obtain a closed form for the prepotential as a function of a. Finally, we show that ∂τ^τ^=1/8πib1<φ>2τ^, where b1 is the one-loop coefficient of the beta function