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

  • $\begingroup$ What's $x$ in the first two equations? $\endgroup$ – Andrei Dec 17 '18 at 4:58
  • 1
    $\begingroup$ @Andrei i have edits its $\endgroup$ – jasmine Dec 17 '18 at 4:59
  • 2
    $\begingroup$ 3,4 do not contradict Liouville $\endgroup$ – zhw. Dec 17 '18 at 5:12
  • $\begingroup$ 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 $|z|\le\sqrt[19]{2+R}$. $\endgroup$ – LutzL Dec 17 '18 at 17:45
  • $\begingroup$ I understood the explanations of 1,2,3. Why (4) is false? @zhw $\endgroup$ – Unknown x Jun 15 at 2:14

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