Koebe 1/4 theorem and inequalities in N=2 supersymmetric QCD

The critical curve C which Im<(tau)over cap>=0, <(tau)over cap>=a(D)/a, in, determines hyperbolic domains whose Poincare metric can be constructed in terms of a(D) 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, a(D), 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 partial derivative(<(tau)over cap>) [tr phi(2)]<((tau)over cap>)=1/8 pi ib(1)[phi](<(tau)over cap>)(2), where b(1) is the one-loop coefficient of the beta function.