Which of the following statements are true..?












1












$begingroup$


Consider the entire function $f(z)=1+z+z^{20}$ and $g(z)=e^z,zin mathbb{C}$ 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}({zin mathbb{C}:|z|le R})$ is bounded for every $R>0$



$4)$$g^{-1}({zin 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






complex-analysis







share|cite|improve this question











$endgroup$












  • $begingroup$
    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 $|z|lesqrt[19]{2+R}$.
    $endgroup$
    – LutzL
    Dec 17 '18 at 17:45
















1












$begingroup$


Consider the entire function $f(z)=1+z+z^{20}$ and $g(z)=e^z,zin mathbb{C}$ 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}({zin mathbb{C}:|z|le R})$ is bounded for every $R>0$



$4)$$g^{-1}({zin 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






complex-analysis







share|cite|improve this question











$endgroup$












  • $begingroup$
    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 $|z|lesqrt[19]{2+R}$.
    $endgroup$
    – LutzL
    Dec 17 '18 at 17:45














1












1








1





$begingroup$


Consider the entire function $f(z)=1+z+z^{20}$ and $g(z)=e^z,zin mathbb{C}$ 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}({zin mathbb{C}:|z|le R})$ is bounded for every $R>0$



$4)$$g^{-1}({zin 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






complex-analysis







share|cite|improve this question











$endgroup$




Consider the entire function $f(z)=1+z+z^{20}$ and $g(z)=e^z,zin mathbb{C}$ 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}({zin mathbb{C}:|z|le R})$ is bounded for every $R>0$



$4)$$g^{-1}({zin 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






complex-analysis




complex-analysis






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 17 '18 at 9:40







jasmine

















asked Dec 17 '18 at 4:55









jasminejasmine

1,791417




1,791417












  • $begingroup$
    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 $|z|lesqrt[19]{2+R}$.
    $endgroup$
    – LutzL
    Dec 17 '18 at 17:45


















  • $begingroup$
    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 $|z|lesqrt[19]{2+R}$.
    $endgroup$
    – LutzL
    Dec 17 '18 at 17:45
















$begingroup$
What's $x$ in the first two equations?
$endgroup$
– Andrei
Dec 17 '18 at 4:58




$begingroup$
What's $x$ in the first two equations?
$endgroup$
– Andrei
Dec 17 '18 at 4:58




1




1




$begingroup$
@Andrei i have edits its
$endgroup$
– jasmine
Dec 17 '18 at 4:59




$begingroup$
@Andrei i have edits its
$endgroup$
– jasmine
Dec 17 '18 at 4:59




2




2




$begingroup$
3,4 do not contradict Liouville
$endgroup$
– zhw.
Dec 17 '18 at 5:12




$begingroup$
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 $|z|lesqrt[19]{2+R}$.
$endgroup$
– LutzL
Dec 17 '18 at 17:45




$begingroup$
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 $|z|lesqrt[19]{2+R}$.
$endgroup$
– LutzL
Dec 17 '18 at 17:45










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