Does there exist a map \(f: \DD\to \DD\) with
\(f\qty{1\over 2} = {3\over 4}\)
\(f'\qty{1\over 2} = {2\over 3}\)
Solution
Apply Schwarz-Pick: \[ \abs{f'\qty{1\over 2}} \leq {1 - \abs{f\qty{1\over 2}}^2 \over 1 - \abs{1\over 2}^2 } = {1-(3/4)^2\over 1-(1/2)^2} = {7/16\over 3/4} = {7\over 12}. \] Since \(7/12<2/3\), the prescribed derivative is impossible.