Elementary-Looking Inequality on n Complex Numbers
$begingroup$
Let $z_1,z_2,ldots,z_n$ be complex numbers. Is it true that $displaystylesum_{1le i,jle n} |z_i+z_j| ge displaystylesum_{1le i,jle n} |z_i-z_j|$?
I know the inequality holds for reals and probably complex numbers... Unfortunately I do not have a proof of either scenario.
This seems like it would fall to some kind of induction and triangle inequalities but I have tried without too much success.
A proof of the inequality for $z_iin mathbb R$ or $z_iin mathbb C$ would be appreciated.
Note: The sigma includes when $i<j,i=j, i>j$ so please keep this in mind.
inequality complex-numbers
edited Feb 15 '16 at 17:38
Michael Hardy
1
asked Feb 15 '16 at 16:56
James LiJames Li
968
$endgroup$
add a comment |
$begingroup$
Let $z_1,z_2,ldots,z_n$ be complex numbers. Is it true that $displaystylesum_{1le i,jle n} |z_i+z_j| ge displaystylesum_{1le i,jle n} |z_i-z_j|$?
I know the inequality holds for reals and probably complex numbers... Unfortunately I do not have a proof of either scenario.
This seems like it would fall to some kind of induction and triangle inequalities but I have tried without too much success.
A proof of the inequality for $z_iin mathbb R$ or $z_iin mathbb C$ would be appreciated.
Note: The sigma includes when $i<j,i=j, i>j$ so please keep this in mind.
inequality complex-numbers
edited Feb 15 '16 at 17:38
Michael Hardy
1
asked Feb 15 '16 at 16:56
James LiJames Li
968
$endgroup$
$begingroup$
Note that $displaystylesum_{i,j}$ includes when $i=j$ so it still holds :)
$endgroup$
– James Li
Feb 15 '16 at 16:58
$begingroup$
I need more caffeine. Deleting my embarrassing comments...
$endgroup$
– copper.hat
Feb 15 '16 at 17:08
add a comment |
$begingroup$
Let $z_1,z_2,ldots,z_n$ be complex numbers. Is it true that $displaystylesum_{1le i,jle n} |z_i+z_j| ge displaystylesum_{1le i,jle n} |z_i-z_j|$?
I know the inequality holds for reals and probably complex numbers... Unfortunately I do not have a proof of either scenario.
This seems like it would fall to some kind of induction and triangle inequalities but I have tried without too much success.
A proof of the inequality for $z_iin mathbb R$ or $z_iin mathbb C$ would be appreciated.
Note: The sigma includes when $i<j,i=j, i>j$ so please keep this in mind.
inequality complex-numbers
edited Feb 15 '16 at 17:38
Michael Hardy
1
asked Feb 15 '16 at 16:56
James LiJames Li
968
$endgroup$
Let $z_1,z_2,ldots,z_n$ be complex numbers. Is it true that $displaystylesum_{1le i,jle n} |z_i+z_j| ge displaystylesum_{1le i,jle n} |z_i-z_j|$?
I know the inequality holds for reals and probably complex numbers... Unfortunately I do not have a proof of either scenario.
This seems like it would fall to some kind of induction and triangle inequalities but I have tried without too much success.
A proof of the inequality for $z_iin mathbb R$ or $z_iin mathbb C$ would be appreciated.
Note: The sigma includes when $i<j,i=j, i>j$ so please keep this in mind.
inequality complex-numbers
inequality complex-numbers
edited Feb 15 '16 at 17:38
Michael Hardy
1
asked Feb 15 '16 at 16:56
James LiJames Li
968
edited Feb 15 '16 at 17:38
Michael Hardy
1
asked Feb 15 '16 at 16:56
James LiJames Li
968
edited Feb 15 '16 at 17:38
Michael Hardy
1
edited Feb 15 '16 at 17:38
Michael Hardy
1
edited Feb 15 '16 at 17:38
Michael Hardy
1
1
asked Feb 15 '16 at 16:56
James LiJames Li
968
asked Feb 15 '16 at 16:56
James LiJames Li
968
asked Feb 15 '16 at 16:56
James LiJames Li
968
968
$begingroup$
Note that $displaystylesum_{i,j}$ includes when $i=j$ so it still holds :)
$endgroup$
– James Li
Feb 15 '16 at 16:58
$begingroup$
I need more caffeine. Deleting my embarrassing comments...
$endgroup$
– copper.hat
Feb 15 '16 at 17:08
add a comment |
$begingroup$
Note that $displaystylesum_{i,j}$ includes when $i=j$ so it still holds :)
$endgroup$
– James Li
Feb 15 '16 at 16:58
$begingroup$
I need more caffeine. Deleting my embarrassing comments...
$endgroup$
– copper.hat
Feb 15 '16 at 17:08
$begingroup$
Note that $displaystylesum_{i,j}$ includes when $i=j$ so it still holds :)
$endgroup$
– James Li
Feb 15 '16 at 16:58
$begingroup$
Note that $displaystylesum_{i,j}$ includes when $i=j$ so it still holds :)
$endgroup$
– James Li
Feb 15 '16 at 16:58
$begingroup$
I need more caffeine. Deleting my embarrassing comments...
$endgroup$
– copper.hat
Feb 15 '16 at 17:08
$begingroup$
I need more caffeine. Deleting my embarrassing comments...
$endgroup$
– copper.hat
Feb 15 '16 at 17:08
add a comment |
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$begingroup$
Note that $displaystylesum_{i,j}$ includes when $i=j$ so it still holds :)
$endgroup$
– James Li
Feb 15 '16 at 16:58
$begingroup$
I need more caffeine. Deleting my embarrassing comments...
$endgroup$
– copper.hat
Feb 15 '16 at 17:08