Here is the problem: find all functions that are everywhere analytic, have a zero of order two in $z=0$, satisfy the condition |f'(z)|\leq 6|z| and such that $f(i)=-2$. Any hint is welcomed.
Find all analytic functions such that...
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complex-analysis
1 Answers
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Here is a hint: consider f'(z)/z.
Since $f(z)$ has a zero of order two at $z=0$, the derivative f'(z) is also holomorphic, and f'(0)=0. Thus, you may write f'(z) as $z\cdot g(z)$, with $g(z)$ holomorphic. Then, the bound in the statement tells you that $|g(z)|$ is bounded.
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1I wanted to write "You got it" but that's apparently too short for a comment. – 2011-12-09