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 const ructed in terms of a(D) and a. We describe C in a parametric form rela ted to a Schwarzian equation and prove new relations for N=2 supersymm etric SU(2) Yang-Mills theory. In particular, using the Koebe 1/4 theo rem and Schwarz's lemma, we obtain inequalities involving u, a(D), and a which seem related to the renormalization group. Furthermore, we ob tain a closed form for the prepotential as a function of a. Finally, w e 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-lo op coefficient of the beta function.