Let $B$ be an open subset of $C$ and $\partial B $ denotes boundary of $B$

Question

Let $B$ be an open subset of $C$ and $\partial B$ denote the boundary of $B$.Which of the following are correct?

(a) For every entire function $f$ ,we have $\partial(f(B)) \subset f(\partial B)$

(b) For every entire function $f$ and a bounded open set $B$ , we have $\partial(f(B)) \subset f(\partial B)$

(c) For every function $f$ , we have $\partial(f(B)) = f(\partial B)$

(d) There exist an unbounded open subset $B$ of $C$ and an entire function $f$ such that $\partial(f(B)) \subset f(\partial B)$

From where I start, I can't understand.Please help something.

Thank you for your time.

Try $f(z) = 0$, $f(z) = z$, $f(z) = e^z$ and $B= \mathbb{C}$. This should let you answer (a), (c), (d). — copper.hat, May 10 '16 at 05:16
what is the boundary of $B$ here? and you give some particular examples ,how you ensure (a) and (c) are true for every entire function.I think some proof required but I can not start ....thank you@copper-hat — May1, May 10 '16 at 05:49

score 1 · Answer 1 · edited Apr 13 '17 at 12:21

1

Note that $\partial \mathbb{C}= \emptyset$.

(a) Take $f(z) = 0$, $B= \mathbb{C}$.

(c) Take $f(z) = 0$, $B= \mathbb{C}$.

(d) Tahe $f(z) = z$, $B= \mathbb{C}$.

edited Apr 13 '17 at 12:21

Community

answered May 10 '16 at 16:02

copper.hat

For (d) $\partial B=\emptyset$ $f(B)={f(z):z\in B}={z:z\in B}=\mathbb C$. which satisfy containment. How it is false? – Math geek May 21 '19 at 08:26
It is not. Thanks for catching this. Will repair later I think to – copper.hat May 21 '19 at 10:32
...the link for (b) has an actual example. – copper.hat May 21 '19 at 10:33

1 Answers1