1
$\begingroup$

Make an examples $f:\Bbb{C}\to\Bbb{C}$ holomorph (If it exists) with the properties:(Justify your answer)

$|f(z)|\leq \frac{1}{100}$ if $|z|\leq 1$ and

$|f(z)-1|\leq \frac{1}{100}$ if $|z-3|\leq 1$.

My attemp: I confuesd to use Liouville thm or continuity? Please guide me how should I think about it!

$\endgroup$
  • $\begingroup$ If $g$ is an entire function, it is continuous in particular, and hence, bounded by some constant $a = a(D) > 0$ on the disc $D.$ $\endgroup$ – Will M. Jan 9 at 19:57
1
$\begingroup$

Let us define $g(z)=0$ on $|z|\le 1$ and $g(z)=1$ on $|z-3|\le 1$. Then $g$ is analytic on a neigoborhood of $K=\{|z|\le 1\}\cup \{|z-3|\le 1\}$. Note that $\Bbb C\setminus K$ is connected. By Runge's theorem, for every $\epsilon>0$, there exists a polynomial $p(z)$ such that $$ |g(z)-p(z)|\le\epsilon,\quad\forall z\in K. $$ If we let $\epsilon =\frac{1}{100}$, then $p$ is a function we are looking for.

$\endgroup$
0
$\begingroup$

You want an entire function satisfying the given two properties.

What about the constant function $f(z)=1/200$ ?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.