Show that $f$ maps the annulus $G=z.$ into the annulus $ <R_1$

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I was stuck on the following problem:




Let $0 < r_0 < r_1$ and $0 < R_0 < R_1$. Let $G$ be the annulus $z.$ Suppose $f$ is holomorphic on the interior of $G$ and continuous on $partialG$. Further suppose that $f$ has no zeros in $G$ and $|f(z)|=R_i$ for $|z|=r_i$ (for $i=0,1$). Show that $f$ maps $G$ into the annulus $z in mathbbC : R_0 < $.




I really have no idea on how to start. I'd appreciate any hints or solutions even those that rely on more powerful results such as the uniformization theorem.




complex-analysis riemann-surfaces




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  • 1




    Presumably you see how to show $f$ maps $G$ into the disk $|z|<R_1$. Now consider $1/f$....
    – David C. Ullrich
    Jul 21 at 22:49










  • Oh duh, I see how to get it now. We use maximum modulus principle right?
    – user135520
    Jul 21 at 22:58







  • 1




    Right. Complex 101 - if someone suggested the uniformization theorem might be needed they were pulling your leg.
    – David C. Ullrich
    Jul 22 at 0:11














up vote
0
down vote

favorite












I was stuck on the following problem:




Let $0 < r_0 < r_1$ and $0 < R_0 < R_1$. Let $G$ be the annulus $z.$ Suppose $f$ is holomorphic on the interior of $G$ and continuous on $partialG$. Further suppose that $f$ has no zeros in $G$ and $|f(z)|=R_i$ for $|z|=r_i$ (for $i=0,1$). Show that $f$ maps $G$ into the annulus $z in mathbbC : R_0 < $.




I really have no idea on how to start. I'd appreciate any hints or solutions even those that rely on more powerful results such as the uniformization theorem.




complex-analysis riemann-surfaces




share|cite|improve this question















  • 1




    Presumably you see how to show $f$ maps $G$ into the disk $|z|<R_1$. Now consider $1/f$....
    – David C. Ullrich
    Jul 21 at 22:49










  • Oh duh, I see how to get it now. We use maximum modulus principle right?
    – user135520
    Jul 21 at 22:58







  • 1




    Right. Complex 101 - if someone suggested the uniformization theorem might be needed they were pulling your leg.
    – David C. Ullrich
    Jul 22 at 0:11












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I was stuck on the following problem:




Let $0 < r_0 < r_1$ and $0 < R_0 < R_1$. Let $G$ be the annulus $z.$ Suppose $f$ is holomorphic on the interior of $G$ and continuous on $partialG$. Further suppose that $f$ has no zeros in $G$ and $|f(z)|=R_i$ for $|z|=r_i$ (for $i=0,1$). Show that $f$ maps $G$ into the annulus $z in mathbbC : R_0 < $.




I really have no idea on how to start. I'd appreciate any hints or solutions even those that rely on more powerful results such as the uniformization theorem.




complex-analysis riemann-surfaces




share|cite|improve this question











I was stuck on the following problem:




Let $0 < r_0 < r_1$ and $0 < R_0 < R_1$. Let $G$ be the annulus $z.$ Suppose $f$ is holomorphic on the interior of $G$ and continuous on $partialG$. Further suppose that $f$ has no zeros in $G$ and $|f(z)|=R_i$ for $|z|=r_i$ (for $i=0,1$). Show that $f$ maps $G$ into the annulus $z in mathbbC : R_0 < $.




I really have no idea on how to start. I'd appreciate any hints or solutions even those that rely on more powerful results such as the uniformization theorem.





complex-analysis riemann-surfaces





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 21 at 21:35









user135520

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  • 1




    Presumably you see how to show $f$ maps $G$ into the disk $|z|<R_1$. Now consider $1/f$....
    – David C. Ullrich
    Jul 21 at 22:49










  • Oh duh, I see how to get it now. We use maximum modulus principle right?
    – user135520
    Jul 21 at 22:58







  • 1




    Right. Complex 101 - if someone suggested the uniformization theorem might be needed they were pulling your leg.
    – David C. Ullrich
    Jul 22 at 0:11












  • 1




    Presumably you see how to show $f$ maps $G$ into the disk $|z|<R_1$. Now consider $1/f$....
    – David C. Ullrich
    Jul 21 at 22:49










  • Oh duh, I see how to get it now. We use maximum modulus principle right?
    – user135520
    Jul 21 at 22:58







  • 1




    Right. Complex 101 - if someone suggested the uniformization theorem might be needed they were pulling your leg.
    – David C. Ullrich
    Jul 22 at 0:11







1




1




Presumably you see how to show $f$ maps $G$ into the disk $|z|<R_1$. Now consider $1/f$....
– David C. Ullrich
Jul 21 at 22:49




Presumably you see how to show $f$ maps $G$ into the disk $|z|<R_1$. Now consider $1/f$....
– David C. Ullrich
Jul 21 at 22:49












Oh duh, I see how to get it now. We use maximum modulus principle right?
– user135520
Jul 21 at 22:58





Oh duh, I see how to get it now. We use maximum modulus principle right?
– user135520
Jul 21 at 22:58





1




1




Right. Complex 101 - if someone suggested the uniformization theorem might be needed they were pulling your leg.
– David C. Ullrich
Jul 22 at 0:11




Right. Complex 101 - if someone suggested the uniformization theorem might be needed they were pulling your leg.
– David C. Ullrich
Jul 22 at 0:11















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