Show that $left|frac{alpha - beta}{1-bar{alpha}beta}right| < 1$ when $|alpha|,|beta| < 1$
$begingroup$
This is the question I'm stumbling with:
When $|alpha| < 1$ and $|beta| < 1$, show that:
$$left|cfrac{alpha - beta}{1-bar{alpha}beta}right| < 1$$
The chapter that contains this question contains (among others) the triangle inequalities:
$$left||z_1| - |z_2|right| le |z_1 + z_2| le |z_1| + |z_2| $$
I've tried to use the triangle inequalities to increase the dividend and/or decrease the divisor:
$$left|cfrac{alpha - beta}{1-bar{alpha}beta}right| < cfrac{|alpha| +|beta|}{left|1-|bar{alpha}beta|right|}$$
But it's not clear if or why that would be smaller than one. I've also tried to multiply the equation by the conjugated divisor $cfrac{1-alphabar{beta}}{1-alphabar{beta}}$, which gives a real divisor, but the equation does not appear solvable.
Any hint would be much appreciated.
complex-analysis inequality complex-numbers
edited Dec 10 '17 at 9:50
Did
247k23222457
asked Sep 26 '13 at 16:50
AndomarAndomar
286210
$endgroup$
add a comment |
$begingroup$
This is the question I'm stumbling with:
When $|alpha| < 1$ and $|beta| < 1$, show that:
$$left|cfrac{alpha - beta}{1-bar{alpha}beta}right| < 1$$
The chapter that contains this question contains (among others) the triangle inequalities:
$$left||z_1| - |z_2|right| le |z_1 + z_2| le |z_1| + |z_2| $$
I've tried to use the triangle inequalities to increase the dividend and/or decrease the divisor:
$$left|cfrac{alpha - beta}{1-bar{alpha}beta}right| < cfrac{|alpha| +|beta|}{left|1-|bar{alpha}beta|right|}$$
But it's not clear if or why that would be smaller than one. I've also tried to multiply the equation by the conjugated divisor $cfrac{1-alphabar{beta}}{1-alphabar{beta}}$, which gives a real divisor, but the equation does not appear solvable.
Any hint would be much appreciated.
complex-analysis inequality complex-numbers
edited Dec 10 '17 at 9:50
Did
247k23222457
asked Sep 26 '13 at 16:50
AndomarAndomar
286210
$endgroup$
add a comment |
$begingroup$
This is the question I'm stumbling with:
When $|alpha| < 1$ and $|beta| < 1$, show that:
$$left|cfrac{alpha - beta}{1-bar{alpha}beta}right| < 1$$
The chapter that contains this question contains (among others) the triangle inequalities:
$$left||z_1| - |z_2|right| le |z_1 + z_2| le |z_1| + |z_2| $$
I've tried to use the triangle inequalities to increase the dividend and/or decrease the divisor:
$$left|cfrac{alpha - beta}{1-bar{alpha}beta}right| < cfrac{|alpha| +|beta|}{left|1-|bar{alpha}beta|right|}$$
But it's not clear if or why that would be smaller than one. I've also tried to multiply the equation by the conjugated divisor $cfrac{1-alphabar{beta}}{1-alphabar{beta}}$, which gives a real divisor, but the equation does not appear solvable.
Any hint would be much appreciated.
complex-analysis inequality complex-numbers
edited Dec 10 '17 at 9:50
Did
247k23222457
asked Sep 26 '13 at 16:50
AndomarAndomar
286210
$endgroup$
This is the question I'm stumbling with:
When $|alpha| < 1$ and $|beta| < 1$, show that:
$$left|cfrac{alpha - beta}{1-bar{alpha}beta}right| < 1$$
The chapter that contains this question contains (among others) the triangle inequalities:
$$left||z_1| - |z_2|right| le |z_1 + z_2| le |z_1| + |z_2| $$
I've tried to use the triangle inequalities to increase the dividend and/or decrease the divisor:
$$left|cfrac{alpha - beta}{1-bar{alpha}beta}right| < cfrac{|alpha| +|beta|}{left|1-|bar{alpha}beta|right|}$$
But it's not clear if or why that would be smaller than one. I've also tried to multiply the equation by the conjugated divisor $cfrac{1-alphabar{beta}}{1-alphabar{beta}}$, which gives a real divisor, but the equation does not appear solvable.
Any hint would be much appreciated.
complex-analysis inequality complex-numbers
complex-analysis inequality complex-numbers
edited Dec 10 '17 at 9:50
Did
247k23222457
asked Sep 26 '13 at 16:50
AndomarAndomar
286210
edited Dec 10 '17 at 9:50
Did
247k23222457
asked Sep 26 '13 at 16:50
AndomarAndomar
286210
edited Dec 10 '17 at 9:50
Did
247k23222457
edited Dec 10 '17 at 9:50
Did
247k23222457
edited Dec 10 '17 at 9:50
Did
247k23222457
247k23222457
asked Sep 26 '13 at 16:50
AndomarAndomar
286210
asked Sep 26 '13 at 16:50
AndomarAndomar
286210
asked Sep 26 '13 at 16:50
AndomarAndomar
286210
286210
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint:
$$
left| frac{alpha-beta}{1-baralphabeta}right| < 1 Leftrightarrow |alpha-beta|^2 < |1-baralphabeta|^2.
$$
Expand both sides, remembering that $|z|^2 = zbar z$ and simplify. That should get you where you want to be after some algebra.
answered Sep 26 '13 at 16:57
mrfmrf
37.4k54685
$endgroup$
$begingroup$
Interesting, this hint gets me to $|alpha|^2 + |beta|^2 < 1 + |alpha|^2 |beta|^2$. Any hint from there?
$endgroup$
– Andomar
Sep 26 '13 at 18:07
$begingroup$
@Andomar What is $(1-|alpha|^2)(1-|beta|^2)$?
$endgroup$
– mrf
Sep 26 '13 at 18:33
$begingroup$
Very helpful, I like the partial answers! Thanks.
$endgroup$
– Andomar
Sep 26 '13 at 18:44
add a comment |
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1 Answer
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active
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1 Answer
1
active
oldest
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$begingroup$
Hint:
$$
left| frac{alpha-beta}{1-baralphabeta}right| < 1 Leftrightarrow |alpha-beta|^2 < |1-baralphabeta|^2.
$$
Expand both sides, remembering that $|z|^2 = zbar z$ and simplify. That should get you where you want to be after some algebra.
answered Sep 26 '13 at 16:57
mrfmrf
37.4k54685
$endgroup$
$begingroup$
Interesting, this hint gets me to $|alpha|^2 + |beta|^2 < 1 + |alpha|^2 |beta|^2$. Any hint from there?
$endgroup$
– Andomar
Sep 26 '13 at 18:07
$begingroup$
@Andomar What is $(1-|alpha|^2)(1-|beta|^2)$?
$endgroup$
– mrf
Sep 26 '13 at 18:33
$begingroup$
Very helpful, I like the partial answers! Thanks.
$endgroup$
– Andomar
Sep 26 '13 at 18:44
add a comment |
$begingroup$
Hint:
$$
left| frac{alpha-beta}{1-baralphabeta}right| < 1 Leftrightarrow |alpha-beta|^2 < |1-baralphabeta|^2.
$$
Expand both sides, remembering that $|z|^2 = zbar z$ and simplify. That should get you where you want to be after some algebra.
answered Sep 26 '13 at 16:57
mrfmrf
37.4k54685
$endgroup$
$begingroup$
Interesting, this hint gets me to $|alpha|^2 + |beta|^2 < 1 + |alpha|^2 |beta|^2$. Any hint from there?
$endgroup$
– Andomar
Sep 26 '13 at 18:07
$begingroup$
@Andomar What is $(1-|alpha|^2)(1-|beta|^2)$?
$endgroup$
– mrf
Sep 26 '13 at 18:33
$begingroup$
Very helpful, I like the partial answers! Thanks.
$endgroup$
– Andomar
Sep 26 '13 at 18:44
add a comment |
$begingroup$
Hint:
$$
left| frac{alpha-beta}{1-baralphabeta}right| < 1 Leftrightarrow |alpha-beta|^2 < |1-baralphabeta|^2.
$$
Expand both sides, remembering that $|z|^2 = zbar z$ and simplify. That should get you where you want to be after some algebra.
answered Sep 26 '13 at 16:57
mrfmrf
37.4k54685
$endgroup$
Hint:
$$
left| frac{alpha-beta}{1-baralphabeta}right| < 1 Leftrightarrow |alpha-beta|^2 < |1-baralphabeta|^2.
$$
Expand both sides, remembering that $|z|^2 = zbar z$ and simplify. That should get you where you want to be after some algebra.
answered Sep 26 '13 at 16:57
mrfmrf
37.4k54685
answered Sep 26 '13 at 16:57
mrfmrf
37.4k54685
answered Sep 26 '13 at 16:57
mrfmrf
37.4k54685
answered Sep 26 '13 at 16:57
mrfmrf
37.4k54685
37.4k54685
$begingroup$
Interesting, this hint gets me to $|alpha|^2 + |beta|^2 < 1 + |alpha|^2 |beta|^2$. Any hint from there?
$endgroup$
– Andomar
Sep 26 '13 at 18:07
$begingroup$
@Andomar What is $(1-|alpha|^2)(1-|beta|^2)$?
$endgroup$
– mrf
Sep 26 '13 at 18:33
$begingroup$
Very helpful, I like the partial answers! Thanks.
$endgroup$
– Andomar
Sep 26 '13 at 18:44
add a comment |
$begingroup$
Interesting, this hint gets me to $|alpha|^2 + |beta|^2 < 1 + |alpha|^2 |beta|^2$. Any hint from there?
$endgroup$
– Andomar
Sep 26 '13 at 18:07
$begingroup$
@Andomar What is $(1-|alpha|^2)(1-|beta|^2)$?
$endgroup$
– mrf
Sep 26 '13 at 18:33
$begingroup$
Very helpful, I like the partial answers! Thanks.
$endgroup$
– Andomar
Sep 26 '13 at 18:44
$begingroup$
Interesting, this hint gets me to $|alpha|^2 + |beta|^2 < 1 + |alpha|^2 |beta|^2$. Any hint from there?
$endgroup$
– Andomar
Sep 26 '13 at 18:07
$begingroup$
Interesting, this hint gets me to $|alpha|^2 + |beta|^2 < 1 + |alpha|^2 |beta|^2$. Any hint from there?
$endgroup$
– Andomar
Sep 26 '13 at 18:07
$begingroup$
@Andomar What is $(1-|alpha|^2)(1-|beta|^2)$?
$endgroup$
– mrf
Sep 26 '13 at 18:33
$begingroup$
@Andomar What is $(1-|alpha|^2)(1-|beta|^2)$?
$endgroup$
– mrf
Sep 26 '13 at 18:33
$begingroup$
Very helpful, I like the partial answers! Thanks.
$endgroup$
– Andomar
Sep 26 '13 at 18:44
$begingroup$
Very helpful, I like the partial answers! Thanks.
$endgroup$
– Andomar
Sep 26 '13 at 18:44
add a comment |
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