Existance of an analytic function on unit disc
Clash Royale CLAN TAG#URR8PPP
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Is there an analytic function $f:B_1(0)to B_1(0)$ such that $f(0)=1/2$ and $f^prime(0)=3/4$? If it exists, is it unique?
The answer to the first part of the question is affirmative. We can use Scharz Pick lemma to find a Mobius transformation $f$ satisfying the condition. But is the function unique? I got stuck here. Please help.
complex-analysis holomorphic-functions mobius-transformation
add a comment |
up vote
2
down vote
favorite
Is there an analytic function $f:B_1(0)to B_1(0)$ such that $f(0)=1/2$ and $f^prime(0)=3/4$? If it exists, is it unique?
The answer to the first part of the question is affirmative. We can use Scharz Pick lemma to find a Mobius transformation $f$ satisfying the condition. But is the function unique? I got stuck here. Please help.
complex-analysis holomorphic-functions mobius-transformation
here $B_1(0)$ is the open ball or radius 1 centered at zero, or there is something different?
– Masacroso
yesterday
1
@Anupam You may be interested in knowing that 3/4 is the largest possible value of $|f'(0)|$. You can look for "An extremal problem" in Rudin's RCA (chapter on MMP).
– Kavi Rama Murthy
yesterday
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Is there an analytic function $f:B_1(0)to B_1(0)$ such that $f(0)=1/2$ and $f^prime(0)=3/4$? If it exists, is it unique?
The answer to the first part of the question is affirmative. We can use Scharz Pick lemma to find a Mobius transformation $f$ satisfying the condition. But is the function unique? I got stuck here. Please help.
complex-analysis holomorphic-functions mobius-transformation
Is there an analytic function $f:B_1(0)to B_1(0)$ such that $f(0)=1/2$ and $f^prime(0)=3/4$? If it exists, is it unique?
The answer to the first part of the question is affirmative. We can use Scharz Pick lemma to find a Mobius transformation $f$ satisfying the condition. But is the function unique? I got stuck here. Please help.
complex-analysis holomorphic-functions mobius-transformation
complex-analysis holomorphic-functions mobius-transformation
asked yesterday
Anupam
2,2381823
2,2381823
here $B_1(0)$ is the open ball or radius 1 centered at zero, or there is something different?
– Masacroso
yesterday
1
@Anupam You may be interested in knowing that 3/4 is the largest possible value of $|f'(0)|$. You can look for "An extremal problem" in Rudin's RCA (chapter on MMP).
– Kavi Rama Murthy
yesterday
add a comment |
here $B_1(0)$ is the open ball or radius 1 centered at zero, or there is something different?
– Masacroso
yesterday
1
@Anupam You may be interested in knowing that 3/4 is the largest possible value of $|f'(0)|$. You can look for "An extremal problem" in Rudin's RCA (chapter on MMP).
– Kavi Rama Murthy
yesterday
here $B_1(0)$ is the open ball or radius 1 centered at zero, or there is something different?
– Masacroso
yesterday
here $B_1(0)$ is the open ball or radius 1 centered at zero, or there is something different?
– Masacroso
yesterday
1
1
@Anupam You may be interested in knowing that 3/4 is the largest possible value of $|f'(0)|$. You can look for "An extremal problem" in Rudin's RCA (chapter on MMP).
– Kavi Rama Murthy
yesterday
@Anupam You may be interested in knowing that 3/4 is the largest possible value of $|f'(0)|$. You can look for "An extremal problem" in Rudin's RCA (chapter on MMP).
– Kavi Rama Murthy
yesterday
add a comment |
3 Answers
3
active
oldest
votes
up vote
4
down vote
accepted
$f(z)=frac 2z+1 z+2$ is one function with these properties. If $g$ is another such function define $h(z)=phi (g(z))$ where $phi (z)=frac z-frac 1 2 1-frac 1 2 z$. You can verify that $h$ maps $B(0,1)$ into itself and vanishes at $0$. A simple calculation shows $h'(0)=1$. By Schwarz Lemma we get $h(z)=z$ for all $z$ from which we get $g(z)=f(z)$.
@ Kavi Rama Murthty: That's so nice. Thanks. Can we say anything on existence and uniqueness of an analytic function from open unit disc to open unit disc fatisfying $f(0)=1/2, f^prime(0)=3/8?$. Here the problem is that equality in Schawrz-Pick lemma does not hold.
– Anupam
yesterday
@Anupam $frac 3z 8 +frac 1 2$ and $frac 1 2 e^frac 3 4 z$ are two solutions in this case.
– Kavi Rama Murthy
yesterday
@ Kavi Rama Murthy: Is there any standard way to find the solutions?
– Anupam
yesterday
@ Kavi Rama Murthy: Will $1/2e^frac34z$ keep images from open unit disc to open unit disc?
– Anupam
yesterday
@Anupam I am sorry, it does not. My example is wrong. There is no general method for finding such functions.
– Kavi Rama Murthy
21 hours ago
add a comment |
up vote
2
down vote
By Schwarz-Pick Lemma, for all $|z|<1$,
$$|f'(z)|leq frac1-1-.$$
Note that if equality holds at some point then
$f$ must be an analytic automorphism of the unit disc.
Since for $z=0$ the above equality holds then
$$f(z)=e^ithetafracz-a1-baraz$$
with $|a|<1$ and $thetainmathbbR$.
Now
$$
begincasesf(0)=-e^ithetaa=1/2\
f'(0)=e^itheta(1-|a|^2)=3/4
endcases
Leftrightarrow
begincasesa=-1/2\
e^itheta=1
endcases
$$
and we may conclude that $f$ exists and it is unique:
$$f(z)=frac2z+1z+2.$$
add a comment |
up vote
1
down vote
The Mobius tfm $zmapsto frac1+2z2+z$ does the job. If you want $f$ to be a bijection, then the answer is yes because the only transforms that map the disk into itself bijectively are $zmapsto fracz-a1-overlineaze^itheta$, and $a,theta$ are determined by the conditions. There is something special about the choice $f'(0)=frac34$, because otherwise you could choose $zmapsto fracz-a1-overlineazrho$ with $|rho|<1$.
Kavi's answer was better than this
– Richard Martin
yesterday
add a comment |
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
$f(z)=frac 2z+1 z+2$ is one function with these properties. If $g$ is another such function define $h(z)=phi (g(z))$ where $phi (z)=frac z-frac 1 2 1-frac 1 2 z$. You can verify that $h$ maps $B(0,1)$ into itself and vanishes at $0$. A simple calculation shows $h'(0)=1$. By Schwarz Lemma we get $h(z)=z$ for all $z$ from which we get $g(z)=f(z)$.
@ Kavi Rama Murthty: That's so nice. Thanks. Can we say anything on existence and uniqueness of an analytic function from open unit disc to open unit disc fatisfying $f(0)=1/2, f^prime(0)=3/8?$. Here the problem is that equality in Schawrz-Pick lemma does not hold.
– Anupam
yesterday
@Anupam $frac 3z 8 +frac 1 2$ and $frac 1 2 e^frac 3 4 z$ are two solutions in this case.
– Kavi Rama Murthy
yesterday
@ Kavi Rama Murthy: Is there any standard way to find the solutions?
– Anupam
yesterday
@ Kavi Rama Murthy: Will $1/2e^frac34z$ keep images from open unit disc to open unit disc?
– Anupam
yesterday
@Anupam I am sorry, it does not. My example is wrong. There is no general method for finding such functions.
– Kavi Rama Murthy
21 hours ago
add a comment |
up vote
4
down vote
accepted
$f(z)=frac 2z+1 z+2$ is one function with these properties. If $g$ is another such function define $h(z)=phi (g(z))$ where $phi (z)=frac z-frac 1 2 1-frac 1 2 z$. You can verify that $h$ maps $B(0,1)$ into itself and vanishes at $0$. A simple calculation shows $h'(0)=1$. By Schwarz Lemma we get $h(z)=z$ for all $z$ from which we get $g(z)=f(z)$.
@ Kavi Rama Murthty: That's so nice. Thanks. Can we say anything on existence and uniqueness of an analytic function from open unit disc to open unit disc fatisfying $f(0)=1/2, f^prime(0)=3/8?$. Here the problem is that equality in Schawrz-Pick lemma does not hold.
– Anupam
yesterday
@Anupam $frac 3z 8 +frac 1 2$ and $frac 1 2 e^frac 3 4 z$ are two solutions in this case.
– Kavi Rama Murthy
yesterday
@ Kavi Rama Murthy: Is there any standard way to find the solutions?
– Anupam
yesterday
@ Kavi Rama Murthy: Will $1/2e^frac34z$ keep images from open unit disc to open unit disc?
– Anupam
yesterday
@Anupam I am sorry, it does not. My example is wrong. There is no general method for finding such functions.
– Kavi Rama Murthy
21 hours ago
add a comment |
up vote
4
down vote
accepted
up vote
4
down vote
accepted
$f(z)=frac 2z+1 z+2$ is one function with these properties. If $g$ is another such function define $h(z)=phi (g(z))$ where $phi (z)=frac z-frac 1 2 1-frac 1 2 z$. You can verify that $h$ maps $B(0,1)$ into itself and vanishes at $0$. A simple calculation shows $h'(0)=1$. By Schwarz Lemma we get $h(z)=z$ for all $z$ from which we get $g(z)=f(z)$.
$f(z)=frac 2z+1 z+2$ is one function with these properties. If $g$ is another such function define $h(z)=phi (g(z))$ where $phi (z)=frac z-frac 1 2 1-frac 1 2 z$. You can verify that $h$ maps $B(0,1)$ into itself and vanishes at $0$. A simple calculation shows $h'(0)=1$. By Schwarz Lemma we get $h(z)=z$ for all $z$ from which we get $g(z)=f(z)$.
answered yesterday
Kavi Rama Murthy
39.1k31748
39.1k31748
@ Kavi Rama Murthty: That's so nice. Thanks. Can we say anything on existence and uniqueness of an analytic function from open unit disc to open unit disc fatisfying $f(0)=1/2, f^prime(0)=3/8?$. Here the problem is that equality in Schawrz-Pick lemma does not hold.
– Anupam
yesterday
@Anupam $frac 3z 8 +frac 1 2$ and $frac 1 2 e^frac 3 4 z$ are two solutions in this case.
– Kavi Rama Murthy
yesterday
@ Kavi Rama Murthy: Is there any standard way to find the solutions?
– Anupam
yesterday
@ Kavi Rama Murthy: Will $1/2e^frac34z$ keep images from open unit disc to open unit disc?
– Anupam
yesterday
@Anupam I am sorry, it does not. My example is wrong. There is no general method for finding such functions.
– Kavi Rama Murthy
21 hours ago
add a comment |
@ Kavi Rama Murthty: That's so nice. Thanks. Can we say anything on existence and uniqueness of an analytic function from open unit disc to open unit disc fatisfying $f(0)=1/2, f^prime(0)=3/8?$. Here the problem is that equality in Schawrz-Pick lemma does not hold.
– Anupam
yesterday
@Anupam $frac 3z 8 +frac 1 2$ and $frac 1 2 e^frac 3 4 z$ are two solutions in this case.
– Kavi Rama Murthy
yesterday
@ Kavi Rama Murthy: Is there any standard way to find the solutions?
– Anupam
yesterday
@ Kavi Rama Murthy: Will $1/2e^frac34z$ keep images from open unit disc to open unit disc?
– Anupam
yesterday
@Anupam I am sorry, it does not. My example is wrong. There is no general method for finding such functions.
– Kavi Rama Murthy
21 hours ago
@ Kavi Rama Murthty: That's so nice. Thanks. Can we say anything on existence and uniqueness of an analytic function from open unit disc to open unit disc fatisfying $f(0)=1/2, f^prime(0)=3/8?$. Here the problem is that equality in Schawrz-Pick lemma does not hold.
– Anupam
yesterday
@ Kavi Rama Murthty: That's so nice. Thanks. Can we say anything on existence and uniqueness of an analytic function from open unit disc to open unit disc fatisfying $f(0)=1/2, f^prime(0)=3/8?$. Here the problem is that equality in Schawrz-Pick lemma does not hold.
– Anupam
yesterday
@Anupam $frac 3z 8 +frac 1 2$ and $frac 1 2 e^frac 3 4 z$ are two solutions in this case.
– Kavi Rama Murthy
yesterday
@Anupam $frac 3z 8 +frac 1 2$ and $frac 1 2 e^frac 3 4 z$ are two solutions in this case.
– Kavi Rama Murthy
yesterday
@ Kavi Rama Murthy: Is there any standard way to find the solutions?
– Anupam
yesterday
@ Kavi Rama Murthy: Is there any standard way to find the solutions?
– Anupam
yesterday
@ Kavi Rama Murthy: Will $1/2e^frac34z$ keep images from open unit disc to open unit disc?
– Anupam
yesterday
@ Kavi Rama Murthy: Will $1/2e^frac34z$ keep images from open unit disc to open unit disc?
– Anupam
yesterday
@Anupam I am sorry, it does not. My example is wrong. There is no general method for finding such functions.
– Kavi Rama Murthy
21 hours ago
@Anupam I am sorry, it does not. My example is wrong. There is no general method for finding such functions.
– Kavi Rama Murthy
21 hours ago
add a comment |
up vote
2
down vote
By Schwarz-Pick Lemma, for all $|z|<1$,
$$|f'(z)|leq frac1-1-.$$
Note that if equality holds at some point then
$f$ must be an analytic automorphism of the unit disc.
Since for $z=0$ the above equality holds then
$$f(z)=e^ithetafracz-a1-baraz$$
with $|a|<1$ and $thetainmathbbR$.
Now
$$
begincasesf(0)=-e^ithetaa=1/2\
f'(0)=e^itheta(1-|a|^2)=3/4
endcases
Leftrightarrow
begincasesa=-1/2\
e^itheta=1
endcases
$$
and we may conclude that $f$ exists and it is unique:
$$f(z)=frac2z+1z+2.$$
add a comment |
up vote
2
down vote
By Schwarz-Pick Lemma, for all $|z|<1$,
$$|f'(z)|leq frac1-1-.$$
Note that if equality holds at some point then
$f$ must be an analytic automorphism of the unit disc.
Since for $z=0$ the above equality holds then
$$f(z)=e^ithetafracz-a1-baraz$$
with $|a|<1$ and $thetainmathbbR$.
Now
$$
begincasesf(0)=-e^ithetaa=1/2\
f'(0)=e^itheta(1-|a|^2)=3/4
endcases
Leftrightarrow
begincasesa=-1/2\
e^itheta=1
endcases
$$
and we may conclude that $f$ exists and it is unique:
$$f(z)=frac2z+1z+2.$$
add a comment |
up vote
2
down vote
up vote
2
down vote
By Schwarz-Pick Lemma, for all $|z|<1$,
$$|f'(z)|leq frac1-1-.$$
Note that if equality holds at some point then
$f$ must be an analytic automorphism of the unit disc.
Since for $z=0$ the above equality holds then
$$f(z)=e^ithetafracz-a1-baraz$$
with $|a|<1$ and $thetainmathbbR$.
Now
$$
begincasesf(0)=-e^ithetaa=1/2\
f'(0)=e^itheta(1-|a|^2)=3/4
endcases
Leftrightarrow
begincasesa=-1/2\
e^itheta=1
endcases
$$
and we may conclude that $f$ exists and it is unique:
$$f(z)=frac2z+1z+2.$$
By Schwarz-Pick Lemma, for all $|z|<1$,
$$|f'(z)|leq frac1-1-.$$
Note that if equality holds at some point then
$f$ must be an analytic automorphism of the unit disc.
Since for $z=0$ the above equality holds then
$$f(z)=e^ithetafracz-a1-baraz$$
with $|a|<1$ and $thetainmathbbR$.
Now
$$
begincasesf(0)=-e^ithetaa=1/2\
f'(0)=e^itheta(1-|a|^2)=3/4
endcases
Leftrightarrow
begincasesa=-1/2\
e^itheta=1
endcases
$$
and we may conclude that $f$ exists and it is unique:
$$f(z)=frac2z+1z+2.$$
edited yesterday
answered yesterday
Robert Z
89.3k1056128
89.3k1056128
add a comment |
add a comment |
up vote
1
down vote
The Mobius tfm $zmapsto frac1+2z2+z$ does the job. If you want $f$ to be a bijection, then the answer is yes because the only transforms that map the disk into itself bijectively are $zmapsto fracz-a1-overlineaze^itheta$, and $a,theta$ are determined by the conditions. There is something special about the choice $f'(0)=frac34$, because otherwise you could choose $zmapsto fracz-a1-overlineazrho$ with $|rho|<1$.
Kavi's answer was better than this
– Richard Martin
yesterday
add a comment |
up vote
1
down vote
The Mobius tfm $zmapsto frac1+2z2+z$ does the job. If you want $f$ to be a bijection, then the answer is yes because the only transforms that map the disk into itself bijectively are $zmapsto fracz-a1-overlineaze^itheta$, and $a,theta$ are determined by the conditions. There is something special about the choice $f'(0)=frac34$, because otherwise you could choose $zmapsto fracz-a1-overlineazrho$ with $|rho|<1$.
Kavi's answer was better than this
– Richard Martin
yesterday
add a comment |
up vote
1
down vote
up vote
1
down vote
The Mobius tfm $zmapsto frac1+2z2+z$ does the job. If you want $f$ to be a bijection, then the answer is yes because the only transforms that map the disk into itself bijectively are $zmapsto fracz-a1-overlineaze^itheta$, and $a,theta$ are determined by the conditions. There is something special about the choice $f'(0)=frac34$, because otherwise you could choose $zmapsto fracz-a1-overlineazrho$ with $|rho|<1$.
The Mobius tfm $zmapsto frac1+2z2+z$ does the job. If you want $f$ to be a bijection, then the answer is yes because the only transforms that map the disk into itself bijectively are $zmapsto fracz-a1-overlineaze^itheta$, and $a,theta$ are determined by the conditions. There is something special about the choice $f'(0)=frac34$, because otherwise you could choose $zmapsto fracz-a1-overlineazrho$ with $|rho|<1$.
edited yesterday
answered yesterday
Richard Martin
9648
9648
Kavi's answer was better than this
– Richard Martin
yesterday
add a comment |
Kavi's answer was better than this
– Richard Martin
yesterday
Kavi's answer was better than this
– Richard Martin
yesterday
Kavi's answer was better than this
– Richard Martin
yesterday
add a comment |
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here $B_1(0)$ is the open ball or radius 1 centered at zero, or there is something different?
– Masacroso
yesterday
1
@Anupam You may be interested in knowing that 3/4 is the largest possible value of $|f'(0)|$. You can look for "An extremal problem" in Rudin's RCA (chapter on MMP).
– Kavi Rama Murthy
yesterday