2 problems in complex variable

**fuzzy topology** · February 12th 2010, 10:07 AM

Welcome my dear :

I need your help in solving these 2 problems in complex variable. You will find my
questions in the file attached.

Please give me an idea and I will try to proceed.

Thank you in advance

**fuzzy topology** · February 12th 2010, 10:30 AM

I get the proof for the second question and what I want know is just the proof of Cauchy's inequality by induction

**Drexel28** · February 12th 2010, 12:27 PM

Originally Posted by fuzzy topology

Welcome my dear :

I need your help in solving these 2 problems in complex variable. You will find my
questions in the file attached.

Please give me an idea and I will try to proceed.

Thank you in advance

It's clear for the first case. Now suppose that $\left|\sum_{i=1}^n a_ib_i\right|^2\leqslant \sum_{i=1}^n|a_i|^2\sum_{i=1}^n|b_i|^2$ then $\left|\sum_{i=1}^{n+1}a_ib_i\right|^2=\left|\sum_{ i=1}^{n}a_ib_i+a_nb_n\right|^2\leqslant\cdots$ .

Finish.

**Random Variable** · February 12th 2010, 01:20 PM

You probably have alread realized this but $1 + \cos \theta + \cos 2 \theta + \cos 3 \theta + ... + \cos n \theta= \frac{1}{2} + \frac{1}{2} \sum_{k=-n}^{n} e^{ik \theta}$

**fuzzy topology** · February 12th 2010, 09:07 PM

Thank you Drexel28 very much, I know understand the concept.

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