complex analysis : Growth
$begingroup$
Let $f$ holomorphic on $C$.
I'm looking for a counter exemple to : If $sup_{|z|=r} |Re(f)| = O(r^{d})$ then $sup_{|z|=r} |f| = O(r^{d})$
Actually, I'm wondering if I can find a entire function such as the growth of the real part is lower than the growth of the imaginary part.
Thank you for reading me :).
complex-analysis entire-functions
asked Jan 5 at 18:03
CechMSCechMS
447
$endgroup$
add a comment |
$begingroup$
Let $f$ holomorphic on $C$.
I'm looking for a counter exemple to : If $sup_{|z|=r} |Re(f)| = O(r^{d})$ then $sup_{|z|=r} |f| = O(r^{d})$
Actually, I'm wondering if I can find a entire function such as the growth of the real part is lower than the growth of the imaginary part.
Thank you for reading me :).
complex-analysis entire-functions
asked Jan 5 at 18:03
CechMSCechMS
447
$endgroup$
add a comment |
$begingroup$
Let $f$ holomorphic on $C$.
I'm looking for a counter exemple to : If $sup_{|z|=r} |Re(f)| = O(r^{d})$ then $sup_{|z|=r} |f| = O(r^{d})$
Actually, I'm wondering if I can find a entire function such as the growth of the real part is lower than the growth of the imaginary part.
Thank you for reading me :).
complex-analysis entire-functions
asked Jan 5 at 18:03
CechMSCechMS
447
$endgroup$
Let $f$ holomorphic on $C$.
I'm looking for a counter exemple to : If $sup_{|z|=r} |Re(f)| = O(r^{d})$ then $sup_{|z|=r} |f| = O(r^{d})$
Actually, I'm wondering if I can find a entire function such as the growth of the real part is lower than the growth of the imaginary part.
Thank you for reading me :).
complex-analysis entire-functions
complex-analysis entire-functions
asked Jan 5 at 18:03
CechMSCechMS
447
asked Jan 5 at 18:03
CechMSCechMS
447
asked Jan 5 at 18:03
CechMSCechMS
447
asked Jan 5 at 18:03
CechMSCechMS
447
asked Jan 5 at 18:03
CechMSCechMS
447
447
add a comment |
add a comment |
1 Answer
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$begingroup$
It is a beautiful fact about entire functions that there is no counterexample to your claim! This is the Borel—Carathéodory theorem. (In the formulation on that web page, take $R=2r$ to deduce the exact form of your claim.)
answered Jan 5 at 18:15
Greg MartinGreg Martin
36.4k23565
$endgroup$
add a comment |
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1 Answer
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$begingroup$
It is a beautiful fact about entire functions that there is no counterexample to your claim! This is the Borel—Carathéodory theorem. (In the formulation on that web page, take $R=2r$ to deduce the exact form of your claim.)
answered Jan 5 at 18:15
Greg MartinGreg Martin
36.4k23565
$endgroup$
add a comment |
$begingroup$
It is a beautiful fact about entire functions that there is no counterexample to your claim! This is the Borel—Carathéodory theorem. (In the formulation on that web page, take $R=2r$ to deduce the exact form of your claim.)
answered Jan 5 at 18:15
Greg MartinGreg Martin
36.4k23565
$endgroup$
add a comment |
$begingroup$
It is a beautiful fact about entire functions that there is no counterexample to your claim! This is the Borel—Carathéodory theorem. (In the formulation on that web page, take $R=2r$ to deduce the exact form of your claim.)
answered Jan 5 at 18:15
Greg MartinGreg Martin
36.4k23565
$endgroup$
It is a beautiful fact about entire functions that there is no counterexample to your claim! This is the Borel—Carathéodory theorem. (In the formulation on that web page, take $R=2r$ to deduce the exact form of your claim.)
answered Jan 5 at 18:15
Greg MartinGreg Martin
36.4k23565
answered Jan 5 at 18:15
Greg MartinGreg Martin
36.4k23565
answered Jan 5 at 18:15
Greg MartinGreg Martin
36.4k23565
answered Jan 5 at 18:15
Greg MartinGreg Martin
36.4k23565
36.4k23565
add a comment |
add a comment |
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