Determine, with motivation, all funtions that is analytic on

{z/ |z| < 4}, with f( 0 ) = i and |f( z ) <= 1 on {z/ |z| < 4}

Printable View

- Mar 19th 2008, 06:18 AMJacquesRouxComplex Analysis
Determine, with motivation, all funtions that is analytic on

{z/ |z| < 4}, with f( 0 ) = i and |f( z ) <= 1 on {z/ |z| < 4} - Mar 19th 2008, 04:00 PMThePerfectHacker
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 $\displaystyle |z|<4$.

- Mar 19th 2008, 11:50 PMJacquesRoux
Thanks for your help!