1
vote
1answer
79 views

Prove that f is one-to-one on D

Let $f$ be an analytic function on a disc $D$ whose center is the point $z_0$. Assume that $|f'(z)-f'(z_0 )|<|f'(z_0)| $ on D. Prove that $f$ is one-to-one on D.
0
votes
1answer
22 views

Maximum Value - Analytic function

I am having a hard time figuring out where to start and what results to use to address the following question: Suppose $f(z)$ is analytic in the unit disc $D=\{z:|z|<1\}$ and continuous in the ...
-1
votes
1answer
148 views

Complex has became so hard after the min\max modulus principle. Need some proofs and examples. [closed]

1) $f(z)$ being non constant and analytic in a domain $D$ if $f(z)$ continuous on $\overline{D}$ and $|f(z)|$ is constant on the boundary I need to prove that $f(z)$ must have a zero inside $D$! 2) ...
0
votes
1answer
44 views

Show that $\sqrt[k]{|z-a_1|\cdots |z-a_k|}$ has a max greater than $R$, and a min less than $R$

This is a homework problem. For $|z| \le R$ and $|a_j| < R$ for $j=1,\ldots, k$, not all zero, show that $\sqrt[k]{|z-a_1|\cdots |z-a_k|}$ has a max greater than $R$, and a min less than $R$. ...