# Bounded polynomials.

• Mar 25th 2013, 12:04 PM
Ant
Bounded polynomials.
Hi,

if $f(z)$ is a polynomial with $z$ a complex number. Then, is it always true that if the domain of $f$ is bounded then $f$ must be a bounded function?

Also, is it aways true that as we increase $|z|$ we see that $|f(z)|$ increase also? I think this must be the case as polynomials only admit non negative powers.

any help or clarification would be appreciate!

Thanks
• Mar 25th 2013, 01:49 PM
xxp9
Re: Bounded polynomials.
1. Yes. |f(z)| = |an z^n +...+ a0 | <= |an||z|^n + ... + |a0|
2. When |z| is big enough this is true. While if z is relatively small consider the example f(z)=1-z. |f(1)|=0<1=|f(0)|