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Does Jensen's Inequality hold for complex numbers?
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We can use Jensen's inequality to show that if $1<p<infty$, then there exists a constant $C>0$ such that for every $x,y in mathbbR$, we have:
$$
|x+y|^p leq C (|x|^p + |y|^p)
$$
Can we show this inequality for complex numbers? If so, can we use Jensen's inequality to show it? According to wikipedia, the usual definition of convexity is only for real vector spaces.
inequality complex-numbers jensen-inequality
asked Jul 27 at 20:46
Sambo
1,2081427
add a comment |Â
up vote
1
down vote
favorite
We can use Jensen's inequality to show that if $1<p<infty$, then there exists a constant $C>0$ such that for every $x,y in mathbbR$, we have:
$$
|x+y|^p leq C (|x|^p + |y|^p)
$$
Can we show this inequality for complex numbers? If so, can we use Jensen's inequality to show it? According to wikipedia, the usual definition of convexity is only for real vector spaces.
inequality complex-numbers jensen-inequality
asked Jul 27 at 20:46
Sambo
1,2081427
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
We can use Jensen's inequality to show that if $1<p<infty$, then there exists a constant $C>0$ such that for every $x,y in mathbbR$, we have:
$$
|x+y|^p leq C (|x|^p + |y|^p)
$$
Can we show this inequality for complex numbers? If so, can we use Jensen's inequality to show it? According to wikipedia, the usual definition of convexity is only for real vector spaces.
inequality complex-numbers jensen-inequality
asked Jul 27 at 20:46
Sambo
1,2081427
We can use Jensen's inequality to show that if $1<p<infty$, then there exists a constant $C>0$ such that for every $x,y in mathbbR$, we have:
$$
|x+y|^p leq C (|x|^p + |y|^p)
$$
Can we show this inequality for complex numbers? If so, can we use Jensen's inequality to show it? According to wikipedia, the usual definition of convexity is only for real vector spaces.
inequality complex-numbers jensen-inequality
asked Jul 27 at 20:46
Sambo
1,2081427
asked Jul 27 at 20:46
Sambo
1,2081427
asked Jul 27 at 20:46
Sambo
1,2081427
asked Jul 27 at 20:46
Sambo
1,2081427
1,2081427
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
For complex numbers $z, w$
$$
|z+w|^p le (|z| + |w|)^p , .
$$
Now apply your inequality to the real numbers $x = |z|$ and
$y = |w|$:
$$
(|z| + |w|)^p le C (|z|^p + |w|^p) , .
$$
answered Jul 27 at 20:55
Martin R
23.8k32743
Of course; it's obvious once you see. Thanks!
– Sambo
Jul 27 at 20:58
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
For complex numbers $z, w$
$$
|z+w|^p le (|z| + |w|)^p , .
$$
Now apply your inequality to the real numbers $x = |z|$ and
$y = |w|$:
$$
(|z| + |w|)^p le C (|z|^p + |w|^p) , .
$$
answered Jul 27 at 20:55
Martin R
23.8k32743
Of course; it's obvious once you see. Thanks!
– Sambo
Jul 27 at 20:58
add a comment |Â
up vote
2
down vote
accepted
For complex numbers $z, w$
$$
|z+w|^p le (|z| + |w|)^p , .
$$
Now apply your inequality to the real numbers $x = |z|$ and
$y = |w|$:
$$
(|z| + |w|)^p le C (|z|^p + |w|^p) , .
$$
answered Jul 27 at 20:55
Martin R
23.8k32743
Of course; it's obvious once you see. Thanks!
– Sambo
Jul 27 at 20:58
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
For complex numbers $z, w$
$$
|z+w|^p le (|z| + |w|)^p , .
$$
Now apply your inequality to the real numbers $x = |z|$ and
$y = |w|$:
$$
(|z| + |w|)^p le C (|z|^p + |w|^p) , .
$$
answered Jul 27 at 20:55
Martin R
23.8k32743
For complex numbers $z, w$
$$
|z+w|^p le (|z| + |w|)^p , .
$$
Now apply your inequality to the real numbers $x = |z|$ and
$y = |w|$:
$$
(|z| + |w|)^p le C (|z|^p + |w|^p) , .
$$
answered Jul 27 at 20:55
Martin R
23.8k32743
answered Jul 27 at 20:55
Martin R
23.8k32743
answered Jul 27 at 20:55
Martin R
23.8k32743
answered Jul 27 at 20:55
Martin R
23.8k32743
23.8k32743
Of course; it's obvious once you see. Thanks!
– Sambo
Jul 27 at 20:58
add a comment |Â
Of course; it's obvious once you see. Thanks!
– Sambo
Jul 27 at 20:58
Of course; it's obvious once you see. Thanks!
– Sambo
Jul 27 at 20:58
Of course; it's obvious once you see. Thanks!
– Sambo
Jul 27 at 20:58
add a comment |Â
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