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Showing that $left|frac1z^4+1right|leqfrac11-r^4$
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I am trying to show that if $|z|=r<1$, then
$$left|frac1z^4+1right|leqfrac11-r^4 (1)$$
I have shown the inequality $$left|frac1z^3+1right|leqfrac11-r^3 (2)$$ holds under the same conditions, but I am having showing the result of $(1)$.
I considered beginalign
|z^4+1|&geqleft||z^4|-|-1|right| \
&=left||z|^4-1right| \
&=left|r^4-1right|
endalign
Now, $r^4-1$ is positive $forall r<1$ $0$ $(rneq -1)$. Ideally, like in $(2)$, if $r^4-1$ was strictly negative then the result would immediately follow.
A hint would be very helpful.
complex-analysis proof-verification inequality
asked Aug 3 at 7:53
Bell
560112
 |Â
show 1 more comment
up vote
2
down vote
favorite
I am trying to show that if $|z|=r<1$, then
$$left|frac1z^4+1right|leqfrac11-r^4 (1)$$
I have shown the inequality $$left|frac1z^3+1right|leqfrac11-r^3 (2)$$ holds under the same conditions, but I am having showing the result of $(1)$.
I considered beginalign
|z^4+1|&geqleft||z^4|-|-1|right| \
&=left||z|^4-1right| \
&=left|r^4-1right|
endalign
Now, $r^4-1$ is positive $forall r<1$ $0$ $(rneq -1)$. Ideally, like in $(2)$, if $r^4-1$ was strictly negative then the result would immediately follow.
A hint would be very helpful.
complex-analysis proof-verification inequality
asked Aug 3 at 7:53
Bell
560112
1
This is the same as saying, that if $|w|<1$, then $|1+w|ge 1-|w|$.
– Lord Shark the Unknown
Aug 3 at 7:54
By the reverse triangle inequality?
– Bell
Aug 3 at 7:56
$|z^4+1|geq 1- |z|^4=1-r^4>0.$
– Riemann
Aug 3 at 7:56
I'm still unclear. It looks like you're using the reverse triangle inequality, which I thought was defined as $left|z_1-z_2right|geqleft||z_1|-|z_2|right|$ for every complex number $z_1, z_2$. It looks as though you have assumed that $|z_1|-|z_2|$, or in this case, $1-r^4$ is positive. I don't think this is the case.
– Bell
Aug 3 at 8:03
1
math.stackexchange.com/questions/2870628/…
– lab bhattacharjee
Aug 3 at 8:13
 |Â
show 1 more comment
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I am trying to show that if $|z|=r<1$, then
$$left|frac1z^4+1right|leqfrac11-r^4 (1)$$
I have shown the inequality $$left|frac1z^3+1right|leqfrac11-r^3 (2)$$ holds under the same conditions, but I am having showing the result of $(1)$.
I considered beginalign
|z^4+1|&geqleft||z^4|-|-1|right| \
&=left||z|^4-1right| \
&=left|r^4-1right|
endalign
Now, $r^4-1$ is positive $forall r<1$ $0$ $(rneq -1)$. Ideally, like in $(2)$, if $r^4-1$ was strictly negative then the result would immediately follow.
A hint would be very helpful.
complex-analysis proof-verification inequality
asked Aug 3 at 7:53
Bell
560112
I am trying to show that if $|z|=r<1$, then
$$left|frac1z^4+1right|leqfrac11-r^4 (1)$$
I have shown the inequality $$left|frac1z^3+1right|leqfrac11-r^3 (2)$$ holds under the same conditions, but I am having showing the result of $(1)$.
I considered beginalign
|z^4+1|&geqleft||z^4|-|-1|right| \
&=left||z|^4-1right| \
&=left|r^4-1right|
endalign
Now, $r^4-1$ is positive $forall r<1$ $0$ $(rneq -1)$. Ideally, like in $(2)$, if $r^4-1$ was strictly negative then the result would immediately follow.
A hint would be very helpful.
complex-analysis proof-verification inequality
asked Aug 3 at 7:53
Bell
560112
asked Aug 3 at 7:53
Bell
560112
asked Aug 3 at 7:53
Bell
560112
asked Aug 3 at 7:53
Bell
560112
560112
1
This is the same as saying, that if $|w|<1$, then $|1+w|ge 1-|w|$.
– Lord Shark the Unknown
Aug 3 at 7:54
By the reverse triangle inequality?
– Bell
Aug 3 at 7:56
$|z^4+1|geq 1- |z|^4=1-r^4>0.$
– Riemann
Aug 3 at 7:56
I'm still unclear. It looks like you're using the reverse triangle inequality, which I thought was defined as $left|z_1-z_2right|geqleft||z_1|-|z_2|right|$ for every complex number $z_1, z_2$. It looks as though you have assumed that $|z_1|-|z_2|$, or in this case, $1-r^4$ is positive. I don't think this is the case.
– Bell
Aug 3 at 8:03
1
math.stackexchange.com/questions/2870628/…
– lab bhattacharjee
Aug 3 at 8:13
 |Â
show 1 more comment
1
This is the same as saying, that if $|w|<1$, then $|1+w|ge 1-|w|$.
– Lord Shark the Unknown
Aug 3 at 7:54
By the reverse triangle inequality?
– Bell
Aug 3 at 7:56
$|z^4+1|geq 1- |z|^4=1-r^4>0.$
– Riemann
Aug 3 at 7:56
I'm still unclear. It looks like you're using the reverse triangle inequality, which I thought was defined as $left|z_1-z_2right|geqleft||z_1|-|z_2|right|$ for every complex number $z_1, z_2$. It looks as though you have assumed that $|z_1|-|z_2|$, or in this case, $1-r^4$ is positive. I don't think this is the case.
– Bell
Aug 3 at 8:03
1
math.stackexchange.com/questions/2870628/…
– lab bhattacharjee
Aug 3 at 8:13
1
1
This is the same as saying, that if $|w|<1$, then $|1+w|ge 1-|w|$.
– Lord Shark the Unknown
Aug 3 at 7:54
This is the same as saying, that if $|w|<1$, then $|1+w|ge 1-|w|$.
– Lord Shark the Unknown
Aug 3 at 7:54
By the reverse triangle inequality?
– Bell
Aug 3 at 7:56
By the reverse triangle inequality?
– Bell
Aug 3 at 7:56
$|z^4+1|geq 1- |z|^4=1-r^4>0.$
– Riemann
Aug 3 at 7:56
$|z^4+1|geq 1- |z|^4=1-r^4>0.$
– Riemann
Aug 3 at 7:56
I'm still unclear. It looks like you're using the reverse triangle inequality, which I thought was defined as $left|z_1-z_2right|geqleft||z_1|-|z_2|right|$ for every complex number $z_1, z_2$. It looks as though you have assumed that $|z_1|-|z_2|$, or in this case, $1-r^4$ is positive. I don't think this is the case.
– Bell
Aug 3 at 8:03
I'm still unclear. It looks like you're using the reverse triangle inequality, which I thought was defined as $left|z_1-z_2right|geqleft||z_1|-|z_2|right|$ for every complex number $z_1, z_2$. It looks as though you have assumed that $|z_1|-|z_2|$, or in this case, $1-r^4$ is positive. I don't think this is the case.
– Bell
Aug 3 at 8:03
1
1
math.stackexchange.com/questions/2870628/…
– lab bhattacharjee
Aug 3 at 8:13
math.stackexchange.com/questions/2870628/…
– lab bhattacharjee
Aug 3 at 8:13
 |Â
show 1 more comment
3 Answers
3
active
oldest
votes
up vote
1
down vote
accepted
Recall that
$$left|frac1z^4+1right|=frac1z^4+1right$$
and since
$$left|z^4+1right|ge left|r^4-1right|=1-r^4ge 0$$
the result follows.
answered Aug 3 at 8:00
gimusi
63.8k73480
How is $1-r^4geq 0$ when $r$ is defined as $r<1$?
– Bell
Aug 3 at 8:04
r is no negative that is $0le r<1$
– gimusi
Aug 3 at 8:08
This is the part I don't understand. I'm unsure of why $r$ cannot be negative? Now that you say it, I agree that the inequality $(1)$ doesn't hold if $r<0$. How did you see this immediately? I overlooked this completely and assumed that $rin(-infty,0)$ as $r<1$.
– Bell
Aug 3 at 8:14
recall that $ r=|z|ge 0$
– gimusi
Aug 3 at 8:15
1
Now it’s clear! Well done. Bye
– gimusi
Aug 3 at 8:18
 |Â
show 1 more comment
up vote
1
down vote
As Lord Shark (indirectly) mentioned, the exponent does not play a role here, as $|z|^n = |z^n|$.
So, it is enough to show that $1-|w| leq |1 + w|$ for $|w| < 1$, which is a direct consequence of $||a|-|b||leq |a-b| stackrelb rightarrow -bLongrightarrow ||a|-|b||leq |a+b|$
answered Aug 3 at 8:29
trancelocation
4,5821413
add a comment |Â
up vote
1
down vote
By the reverse triangle inequality it holds that $|z^4+1| = |1-(-z^4)| geq 1-|z|^4$ and therefore $frac1 leq frac1^4 = frac11-r^4$
answered Aug 3 at 8:03
Andreas Dahlberg
213
Note that the reverse triangle inequality says that $|z-w| geq | |z| - |w| |$ but this implies that $|z-w| geq |z| - |w| $ because if $ |z| - |w| < 0$ then the equality holds trivially since $|z-w| geq 0$.
– Andreas Dahlberg
Aug 3 at 8:30
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Recall that
$$left|frac1z^4+1right|=frac1z^4+1right$$
and since
$$left|z^4+1right|ge left|r^4-1right|=1-r^4ge 0$$
the result follows.
answered Aug 3 at 8:00
gimusi
63.8k73480
How is $1-r^4geq 0$ when $r$ is defined as $r<1$?
– Bell
Aug 3 at 8:04
r is no negative that is $0le r<1$
– gimusi
Aug 3 at 8:08
This is the part I don't understand. I'm unsure of why $r$ cannot be negative? Now that you say it, I agree that the inequality $(1)$ doesn't hold if $r<0$. How did you see this immediately? I overlooked this completely and assumed that $rin(-infty,0)$ as $r<1$.
– Bell
Aug 3 at 8:14
recall that $ r=|z|ge 0$
– gimusi
Aug 3 at 8:15
1
Now it’s clear! Well done. Bye
– gimusi
Aug 3 at 8:18
 |Â
show 1 more comment
up vote
1
down vote
accepted
Recall that
$$left|frac1z^4+1right|=frac1z^4+1right$$
and since
$$left|z^4+1right|ge left|r^4-1right|=1-r^4ge 0$$
the result follows.
answered Aug 3 at 8:00
gimusi
63.8k73480
How is $1-r^4geq 0$ when $r$ is defined as $r<1$?
– Bell
Aug 3 at 8:04
r is no negative that is $0le r<1$
– gimusi
Aug 3 at 8:08
This is the part I don't understand. I'm unsure of why $r$ cannot be negative? Now that you say it, I agree that the inequality $(1)$ doesn't hold if $r<0$. How did you see this immediately? I overlooked this completely and assumed that $rin(-infty,0)$ as $r<1$.
– Bell
Aug 3 at 8:14
recall that $ r=|z|ge 0$
– gimusi
Aug 3 at 8:15
1
Now it’s clear! Well done. Bye
– gimusi
Aug 3 at 8:18
 |Â
show 1 more comment
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Recall that
$$left|frac1z^4+1right|=frac1z^4+1right$$
and since
$$left|z^4+1right|ge left|r^4-1right|=1-r^4ge 0$$
the result follows.
answered Aug 3 at 8:00
gimusi
63.8k73480
Recall that
$$left|frac1z^4+1right|=frac1z^4+1right$$
and since
$$left|z^4+1right|ge left|r^4-1right|=1-r^4ge 0$$
the result follows.
answered Aug 3 at 8:00
gimusi
63.8k73480
answered Aug 3 at 8:00
gimusi
63.8k73480
answered Aug 3 at 8:00
gimusi
63.8k73480
answered Aug 3 at 8:00
gimusi
63.8k73480
63.8k73480
How is $1-r^4geq 0$ when $r$ is defined as $r<1$?
– Bell
Aug 3 at 8:04
r is no negative that is $0le r<1$
– gimusi
Aug 3 at 8:08
This is the part I don't understand. I'm unsure of why $r$ cannot be negative? Now that you say it, I agree that the inequality $(1)$ doesn't hold if $r<0$. How did you see this immediately? I overlooked this completely and assumed that $rin(-infty,0)$ as $r<1$.
– Bell
Aug 3 at 8:14
recall that $ r=|z|ge 0$
– gimusi
Aug 3 at 8:15
1
Now it’s clear! Well done. Bye
– gimusi
Aug 3 at 8:18
 |Â
show 1 more comment
How is $1-r^4geq 0$ when $r$ is defined as $r<1$?
– Bell
Aug 3 at 8:04
r is no negative that is $0le r<1$
– gimusi
Aug 3 at 8:08
This is the part I don't understand. I'm unsure of why $r$ cannot be negative? Now that you say it, I agree that the inequality $(1)$ doesn't hold if $r<0$. How did you see this immediately? I overlooked this completely and assumed that $rin(-infty,0)$ as $r<1$.
– Bell
Aug 3 at 8:14
recall that $ r=|z|ge 0$
– gimusi
Aug 3 at 8:15
1
Now it’s clear! Well done. Bye
– gimusi
Aug 3 at 8:18
How is $1-r^4geq 0$ when $r$ is defined as $r<1$?
– Bell
Aug 3 at 8:04
How is $1-r^4geq 0$ when $r$ is defined as $r<1$?
– Bell
Aug 3 at 8:04
r is no negative that is $0le r<1$
– gimusi
Aug 3 at 8:08
r is no negative that is $0le r<1$
– gimusi
Aug 3 at 8:08
This is the part I don't understand. I'm unsure of why $r$ cannot be negative? Now that you say it, I agree that the inequality $(1)$ doesn't hold if $r<0$. How did you see this immediately? I overlooked this completely and assumed that $rin(-infty,0)$ as $r<1$.
– Bell
Aug 3 at 8:14
This is the part I don't understand. I'm unsure of why $r$ cannot be negative? Now that you say it, I agree that the inequality $(1)$ doesn't hold if $r<0$. How did you see this immediately? I overlooked this completely and assumed that $rin(-infty,0)$ as $r<1$.
– Bell
Aug 3 at 8:14
recall that $ r=|z|ge 0$
– gimusi
Aug 3 at 8:15
recall that $ r=|z|ge 0$
– gimusi
Aug 3 at 8:15
1
1
Now it’s clear! Well done. Bye
– gimusi
Aug 3 at 8:18
Now it’s clear! Well done. Bye
– gimusi
Aug 3 at 8:18
 |Â
show 1 more comment
up vote
1
down vote
As Lord Shark (indirectly) mentioned, the exponent does not play a role here, as $|z|^n = |z^n|$.
So, it is enough to show that $1-|w| leq |1 + w|$ for $|w| < 1$, which is a direct consequence of $||a|-|b||leq |a-b| stackrelb rightarrow -bLongrightarrow ||a|-|b||leq |a+b|$
answered Aug 3 at 8:29
trancelocation
4,5821413
add a comment |Â
up vote
1
down vote
As Lord Shark (indirectly) mentioned, the exponent does not play a role here, as $|z|^n = |z^n|$.
So, it is enough to show that $1-|w| leq |1 + w|$ for $|w| < 1$, which is a direct consequence of $||a|-|b||leq |a-b| stackrelb rightarrow -bLongrightarrow ||a|-|b||leq |a+b|$
answered Aug 3 at 8:29
trancelocation
4,5821413
add a comment |Â
up vote
1
down vote
up vote
1
down vote
As Lord Shark (indirectly) mentioned, the exponent does not play a role here, as $|z|^n = |z^n|$.
So, it is enough to show that $1-|w| leq |1 + w|$ for $|w| < 1$, which is a direct consequence of $||a|-|b||leq |a-b| stackrelb rightarrow -bLongrightarrow ||a|-|b||leq |a+b|$
answered Aug 3 at 8:29
trancelocation
4,5821413
As Lord Shark (indirectly) mentioned, the exponent does not play a role here, as $|z|^n = |z^n|$.
So, it is enough to show that $1-|w| leq |1 + w|$ for $|w| < 1$, which is a direct consequence of $||a|-|b||leq |a-b| stackrelb rightarrow -bLongrightarrow ||a|-|b||leq |a+b|$
answered Aug 3 at 8:29
trancelocation
4,5821413
answered Aug 3 at 8:29
trancelocation
4,5821413
answered Aug 3 at 8:29
trancelocation
4,5821413
answered Aug 3 at 8:29
trancelocation
4,5821413
4,5821413
add a comment |Â
add a comment |Â
up vote
1
down vote
By the reverse triangle inequality it holds that $|z^4+1| = |1-(-z^4)| geq 1-|z|^4$ and therefore $frac1 leq frac1^4 = frac11-r^4$
answered Aug 3 at 8:03
Andreas Dahlberg
213
Note that the reverse triangle inequality says that $|z-w| geq | |z| - |w| |$ but this implies that $|z-w| geq |z| - |w| $ because if $ |z| - |w| < 0$ then the equality holds trivially since $|z-w| geq 0$.
– Andreas Dahlberg
Aug 3 at 8:30
add a comment |Â
up vote
1
down vote
By the reverse triangle inequality it holds that $|z^4+1| = |1-(-z^4)| geq 1-|z|^4$ and therefore $frac1 leq frac1^4 = frac11-r^4$
answered Aug 3 at 8:03
Andreas Dahlberg
213
Note that the reverse triangle inequality says that $|z-w| geq | |z| - |w| |$ but this implies that $|z-w| geq |z| - |w| $ because if $ |z| - |w| < 0$ then the equality holds trivially since $|z-w| geq 0$.
– Andreas Dahlberg
Aug 3 at 8:30
add a comment |Â
up vote
1
down vote
up vote
1
down vote
By the reverse triangle inequality it holds that $|z^4+1| = |1-(-z^4)| geq 1-|z|^4$ and therefore $frac1 leq frac1^4 = frac11-r^4$
answered Aug 3 at 8:03
Andreas Dahlberg
213
By the reverse triangle inequality it holds that $|z^4+1| = |1-(-z^4)| geq 1-|z|^4$ and therefore $frac1 leq frac1^4 = frac11-r^4$
answered Aug 3 at 8:03
Andreas Dahlberg
213
edited Aug 3 at 8:43
answered Aug 3 at 8:03
Andreas Dahlberg
213
answered Aug 3 at 8:03
Andreas Dahlberg
213
answered Aug 3 at 8:03
Andreas Dahlberg
213
213
Note that the reverse triangle inequality says that $|z-w| geq | |z| - |w| |$ but this implies that $|z-w| geq |z| - |w| $ because if $ |z| - |w| < 0$ then the equality holds trivially since $|z-w| geq 0$.
– Andreas Dahlberg
Aug 3 at 8:30
add a comment |Â
Note that the reverse triangle inequality says that $|z-w| geq | |z| - |w| |$ but this implies that $|z-w| geq |z| - |w| $ because if $ |z| - |w| < 0$ then the equality holds trivially since $|z-w| geq 0$.
– Andreas Dahlberg
Aug 3 at 8:30
Note that the reverse triangle inequality says that $|z-w| geq | |z| - |w| |$ but this implies that $|z-w| geq |z| - |w| $ because if $ |z| - |w| < 0$ then the equality holds trivially since $|z-w| geq 0$.
– Andreas Dahlberg
Aug 3 at 8:30
Note that the reverse triangle inequality says that $|z-w| geq | |z| - |w| |$ but this implies that $|z-w| geq |z| - |w| $ because if $ |z| - |w| < 0$ then the equality holds trivially since $|z-w| geq 0$.
– Andreas Dahlberg
Aug 3 at 8:30
add a comment |Â
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1
This is the same as saying, that if $|w|<1$, then $|1+w|ge 1-|w|$.
– Lord Shark the Unknown
Aug 3 at 7:54
By the reverse triangle inequality?
– Bell
Aug 3 at 7:56
$|z^4+1|geq 1- |z|^4=1-r^4>0.$
– Riemann
Aug 3 at 7:56
I'm still unclear. It looks like you're using the reverse triangle inequality, which I thought was defined as $left|z_1-z_2right|geqleft||z_1|-|z_2|right|$ for every complex number $z_1, z_2$. It looks as though you have assumed that $|z_1|-|z_2|$, or in this case, $1-r^4$ is positive. I don't think this is the case.
– Bell
Aug 3 at 8:03
1
math.stackexchange.com/questions/2870628/…
– lab bhattacharjee
Aug 3 at 8:13