Modulus and complex variable

LegendZKiller · **Joined:** Mon, 6 Feb 2012 06:18:16 UTC **Posts:** 25

Let $f:B(0,2)\to B(0,2)$ be an analytic function so that $f(1)=0.$

Prove that for all $z\in B(0,2)$ is $\left| \dfrac{f(z)}{z} \right|\le \left| \dfrac{2(z-1)}{4-z} \right|.$

No clue on this one, any ideas? I thought on considering g(z)=(z-1)f(z)

or perhaps g(z)=zf(z)

but I don't see the trick to solve the problem.

outermeasure · **Posted:** Mon, 6 Feb 2012 06:29:39 UTC

LegendZKiller wrote:

Let $f:B(0,2)\to B(0,2)$ be an analytic function so that $f(1)=0.$

Prove that for all $z\in B(0,2)$ is $\left| \dfrac{f(z)}{z} \right|\le \left| \dfrac{2(z-1)}{4-z} \right|.$

No clue on this one, any ideas? I thought on considering g(z)=(z-1)f(z)

or perhaps g(z)=zf(z)

but I don't see the trick to solve the problem.

Schwarz-Pick.