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How to find a small disk where an analytic function $f(z)$ such that $f(0)=1$ does not have zeros
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Question: Let $f(z)$ be an analytic function in $D_R=leftz:$ such that $|f(z)|<M ,, forall z in D_R$ and $f(0)=1$. Please find $rho in (0,R)$ such that $f(z)$ has no zeros for any $z$ with $|z|le rho$
My thoughts: This is a past exam problem. I suppose I should try to estimate the growth rate of $|f|$, but I meet a lot of difficulty since the information provided is very limited. I guess the answers should be related to $M$ (otherwise it is not necessary to emphasize $|f|$ is bounded), but I do not know what kind of information is implied when we know this upper bound. I have only taken one elementary course in complex variables so maybe there is some important theorems that I'm missing. Thanks for any comments and answers!
complex-analysis
asked 2 days ago
WallTi
103
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up vote
1
down vote
favorite
Question: Let $f(z)$ be an analytic function in $D_R=leftz:$ such that $|f(z)|<M ,, forall z in D_R$ and $f(0)=1$. Please find $rho in (0,R)$ such that $f(z)$ has no zeros for any $z$ with $|z|le rho$
My thoughts: This is a past exam problem. I suppose I should try to estimate the growth rate of $|f|$, but I meet a lot of difficulty since the information provided is very limited. I guess the answers should be related to $M$ (otherwise it is not necessary to emphasize $|f|$ is bounded), but I do not know what kind of information is implied when we know this upper bound. I have only taken one elementary course in complex variables so maybe there is some important theorems that I'm missing. Thanks for any comments and answers!
complex-analysis
asked 2 days ago
WallTi
103
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Question: Let $f(z)$ be an analytic function in $D_R=leftz:$ such that $|f(z)|<M ,, forall z in D_R$ and $f(0)=1$. Please find $rho in (0,R)$ such that $f(z)$ has no zeros for any $z$ with $|z|le rho$
My thoughts: This is a past exam problem. I suppose I should try to estimate the growth rate of $|f|$, but I meet a lot of difficulty since the information provided is very limited. I guess the answers should be related to $M$ (otherwise it is not necessary to emphasize $|f|$ is bounded), but I do not know what kind of information is implied when we know this upper bound. I have only taken one elementary course in complex variables so maybe there is some important theorems that I'm missing. Thanks for any comments and answers!
complex-analysis
asked 2 days ago
WallTi
103
Question: Let $f(z)$ be an analytic function in $D_R=leftz:$ such that $|f(z)|<M ,, forall z in D_R$ and $f(0)=1$. Please find $rho in (0,R)$ such that $f(z)$ has no zeros for any $z$ with $|z|le rho$
My thoughts: This is a past exam problem. I suppose I should try to estimate the growth rate of $|f|$, but I meet a lot of difficulty since the information provided is very limited. I guess the answers should be related to $M$ (otherwise it is not necessary to emphasize $|f|$ is bounded), but I do not know what kind of information is implied when we know this upper bound. I have only taken one elementary course in complex variables so maybe there is some important theorems that I'm missing. Thanks for any comments and answers!
complex-analysis
asked 2 days ago
WallTi
103
asked 2 days ago
WallTi
103
asked 2 days ago
WallTi
103
asked 2 days ago
WallTi
103
103
add a comment |Â
add a comment |Â
1 Answer
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Consider $g(z)=f(z)-1$. Then $|g(z)|<M+1$ on the disc, and it suffices
to find $rho>0$ such that $|g(z)|<1$ for $rho>0$. Let $0<R'<R$
(say $R'=R/2$ if you like). Then $h(z)=g(z)/z$ is holomorphic, and
by maximal modulus theorem, $|h(z)|le (M+1)/R'$ for $|z|le R'$.
Therefore $g(z)le(M+1)|z|/R'$ for $|z|le R'$. Now take $rho$
with $0<rho<R'$ and $(M+1)rho/R'<1$.
answered 2 days ago
Lord Shark the Unknown
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
Consider $g(z)=f(z)-1$. Then $|g(z)|<M+1$ on the disc, and it suffices
to find $rho>0$ such that $|g(z)|<1$ for $rho>0$. Let $0<R'<R$
(say $R'=R/2$ if you like). Then $h(z)=g(z)/z$ is holomorphic, and
by maximal modulus theorem, $|h(z)|le (M+1)/R'$ for $|z|le R'$.
Therefore $g(z)le(M+1)|z|/R'$ for $|z|le R'$. Now take $rho$
with $0<rho<R'$ and $(M+1)rho/R'<1$.
answered 2 days ago
Lord Shark the Unknown
83.9k949111
add a comment |Â
up vote
0
down vote
accepted
Consider $g(z)=f(z)-1$. Then $|g(z)|<M+1$ on the disc, and it suffices
to find $rho>0$ such that $|g(z)|<1$ for $rho>0$. Let $0<R'<R$
(say $R'=R/2$ if you like). Then $h(z)=g(z)/z$ is holomorphic, and
by maximal modulus theorem, $|h(z)|le (M+1)/R'$ for $|z|le R'$.
Therefore $g(z)le(M+1)|z|/R'$ for $|z|le R'$. Now take $rho$
with $0<rho<R'$ and $(M+1)rho/R'<1$.
answered 2 days ago
Lord Shark the Unknown
83.9k949111
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
Consider $g(z)=f(z)-1$. Then $|g(z)|<M+1$ on the disc, and it suffices
to find $rho>0$ such that $|g(z)|<1$ for $rho>0$. Let $0<R'<R$
(say $R'=R/2$ if you like). Then $h(z)=g(z)/z$ is holomorphic, and
by maximal modulus theorem, $|h(z)|le (M+1)/R'$ for $|z|le R'$.
Therefore $g(z)le(M+1)|z|/R'$ for $|z|le R'$. Now take $rho$
with $0<rho<R'$ and $(M+1)rho/R'<1$.
answered 2 days ago
Lord Shark the Unknown
83.9k949111
Consider $g(z)=f(z)-1$. Then $|g(z)|<M+1$ on the disc, and it suffices
to find $rho>0$ such that $|g(z)|<1$ for $rho>0$. Let $0<R'<R$
(say $R'=R/2$ if you like). Then $h(z)=g(z)/z$ is holomorphic, and
by maximal modulus theorem, $|h(z)|le (M+1)/R'$ for $|z|le R'$.
Therefore $g(z)le(M+1)|z|/R'$ for $|z|le R'$. Now take $rho$
with $0<rho<R'$ and $(M+1)rho/R'<1$.
answered 2 days ago
Lord Shark the Unknown
83.9k949111
answered 2 days ago
Lord Shark the Unknown
83.9k949111
answered 2 days ago
Lord Shark the Unknown
83.9k949111
answered 2 days ago
Lord Shark the Unknown
83.9k949111
83.9k949111
add a comment |Â
add a comment |Â
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