# Which of the following statements are true..?

Consider the entire function $$f(z)=1+z+z^{20}$$ and $$g(z)=e^z,z\in \mathbb{C}$$ Which of the following statements are true ?

$$1)\lim_{|z|\to \infty}|f(z)|=\infty$$

$$2)\lim_{|z|\to \infty}|g(z)|=\infty$$

$$3)f^{-1}(\{z\in \mathbb{C}:|z|\le R\})$$ is bounded for every $$R>0$$

$$4)g^{-1}(\{z\in \mathbb{C}:|z|\le R\})$$ is bounded for every $$R>0$$

# My attempt : option $$1$$ is only correct option

option 2) is false $$lim_{x \rightarrow -\infty} e^{-x} = 0$$

option $$3$$ and $$4$$ are contradiction of lioville thorem so it will be false

is its True ??

Any hints/solution will be appreciated

thanks u

• What's $x$ in the first two equations? – Andrei Dec 17 '18 at 4:58
• @Andrei i have edits its – jasmine Dec 17 '18 at 4:59
• 3,4 do not contradict Liouville – zhw. Dec 17 '18 at 5:12
• The existence of root bounds shows that 3) is true, $z^{20}+z+1=w$ implies that $|z|\le 2+|w|=2+R$, with some work $|z|\le\sqrt{2+R}$. – LutzL Dec 17 '18 at 17:45
• I understood the explanations of 1,2,3. Why (4) is false? @zhw – Unknown x Jun 15 at 2:14