Ratio of maximum value of an anlytic function to that of its derivative

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Let $g(z)=(z-w)f(z)$ where $f(z)$ is a non-zero analytic function on and within the unit circle $|z|=1$ with $|w|>1.$ Is it true that
$$displaystylefracmax_=1leqfracf(z)+frac1z-w?$$
If I proceed with
$$displaystylefracmax_=1=fracmax_=1max_=1$$
$$;;;;;;;displaystyleleqfracf(z)max_=1$$
$$leqfracf(z)+frac1z-w,
$$ then whether the last step follows from the immediate previous step? Is there any other alternative method to establish the claim?




complex-analysis




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    Let $g(z)=(z-w)f(z)$ where $f(z)$ is a non-zero analytic function on and within the unit circle $|z|=1$ with $|w|>1.$ Is it true that
    $$displaystylefracmax_=1leqfracf(z)+frac1z-w?$$
    If I proceed with
    $$displaystylefracmax_=1=fracmax_=1max_=1$$
    $$;;;;;;;displaystyleleqfracf(z)max_=1$$
    $$leqfracf(z)+frac1z-w,
    $$ then whether the last step follows from the immediate previous step? Is there any other alternative method to establish the claim?




    complex-analysis




    share|cite|improve this question





















      up vote
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      down vote

      favorite
      1









      up vote
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      down vote

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      Let $g(z)=(z-w)f(z)$ where $f(z)$ is a non-zero analytic function on and within the unit circle $|z|=1$ with $|w|>1.$ Is it true that
      $$displaystylefracmax_=1leqfracf(z)+frac1z-w?$$
      If I proceed with
      $$displaystylefracmax_=1=fracmax_=1max_=1$$
      $$;;;;;;;displaystyleleqfracf(z)max_=1$$
      $$leqfracf(z)+frac1z-w,
      $$ then whether the last step follows from the immediate previous step? Is there any other alternative method to establish the claim?




      complex-analysis




      share|cite|improve this question











      Let $g(z)=(z-w)f(z)$ where $f(z)$ is a non-zero analytic function on and within the unit circle $|z|=1$ with $|w|>1.$ Is it true that
      $$displaystylefracmax_=1leqfracf(z)+frac1z-w?$$
      If I proceed with
      $$displaystylefracmax_=1=fracmax_=1max_=1$$
      $$;;;;;;;displaystyleleqfracf(z)max_=1$$
      $$leqfracf(z)+frac1z-w,
      $$ then whether the last step follows from the immediate previous step? Is there any other alternative method to establish the claim?





      complex-analysis





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









      asked Jul 31 at 12:01









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