Determine, with motivation, all funtions that is analytic on
{z/ |z| < 4}, with f( 0 ) = i and |f( z ) <= 1 on {z/ |z| < 4}
It seems your problem is faulty somehow. Because the way it is stated $\displaystyle f(z) = i$ is the only such mapping. Because if $\displaystyle f$ is non-constant then by the open mapping theorem $\displaystyle f(0)$ has to be a mapped into an interior point of the range of $\displaystyle f$ but that is not possible since $\displaystyle f(0)=i$ is a boundary point. Thus, we conclude that constant functions are the only such functions, thus, $\displaystyle f(z) = i$ for all $\displaystyle |z|<4$.