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The critical curve ${\cal C}$ on which ${\rm Im}\,\hat\tau =0$,$\hat\tau=a_D/a$, determines hyperbolic domains whose Poincar\'e metric isconstructed in terms of $a_D$ and $a$. We describe ${\cal C}$ in a parametricform related to a Schwarzian equation and prove new relations for $N=2$ Super$SU(2)$ Yang-Mills. In particular, using the Koebe 1/4-theorem and Schwarz'slemma, we obtain inequalities involving $u$, $a_D$ and $a$, which seem relatedto the Renormalization Group. Furthermore, we obtain a closed form for theprepotential as function of $a$. Finally, we show that $\partial_{\hat\tau}\langle {\rm tr}\,\phi^2\rangle_{\hat \tau}={1\over 8\pi i b_1}\langle\phi\rangle_{\hat\tau}^2$, where $b_1$ is the one-loop coefficient of the betafunction.Comment: 11 pages, LaTex file, Expanded version: new results, technical  details explained, misprints corrected and references adde

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