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All roots of a complex polynomial have positive imaginary part. Prove that all roots of its derivative also have positive imaginary part.

It's not a homework. This issue has been proposed in the materials to prepare for exams.

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1 Answer 1

If I'm not mistaken, the roots of $f'$ are in the convex hull of the roots of $f$.

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You're right. That's called Gauss-Lucas. And that's a neat application of the partial fraction decomposition of a rational function. –  1015 Apr 2 '13 at 16:06
    
oh? it's so easy and interesting.it's so strange, that this fact wasn't mentioned in the course of complex analysis in my university. thanks –  ekrez Apr 2 '13 at 16:32

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