complex analysis : Growth
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 2 days ago
CechMSCechMS
346
add a comment |
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 2 days ago
CechMSCechMS
346
add a comment |
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 2 days ago
CechMSCechMS
346
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 2 days ago
CechMSCechMS
346
asked 2 days ago
CechMSCechMS
346
asked 2 days ago
CechMSCechMS
346
asked 2 days ago
CechMSCechMS
346
asked 2 days ago
CechMSCechMS
346
346
add a comment |
add a comment |
1 Answer
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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 2 days ago
Greg MartinGreg Martin
34.9k23161
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
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 2 days ago
Greg MartinGreg Martin
34.9k23161
add a comment |
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 2 days ago
Greg MartinGreg Martin
34.9k23161
add a comment |
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 2 days ago
Greg MartinGreg Martin
34.9k23161
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 2 days ago
Greg MartinGreg Martin
34.9k23161
answered 2 days ago
Greg MartinGreg Martin
34.9k23161
answered 2 days ago
Greg MartinGreg Martin
34.9k23161
answered 2 days ago
Greg MartinGreg Martin
34.9k23161
34.9k23161
add a comment |
add a comment |
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