Problem 1787 (difficulty: 7/10)

Let \(\displaystyle w: S(0, 1)\to S(0, 1)\) be regular and let \(\displaystyle |a|<1\). Show that

   a) \(\displaystyle \displaystyle \left|\frac{w(z)-w(a)}{1-\overline{w(a)}w(z)}\right|\le \left|\frac{z-a}{1-\bar{a}z}\right|\)    b) \(\displaystyle \displaystyle |w'(a)|\le\frac{1-|w(a)|^2}{1-|a|^2}\).

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