Thread: 2 problems in complex variable

1. 2 problems in complex variable

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.

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

3. 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.

It's clear for the first case. Now suppose that $\displaystyle \left|\sum_{i=1}^n a_ib_i\right|^2\leqslant \sum_{i=1}^n|a_i|^2\sum_{i=1}^n|b_i|^2$ then $\displaystyle \left|\sum_{i=1}^{n+1}a_ib_i\right|^2=\left|\sum_{ i=1}^{n}a_ib_i+a_nb_n\right|^2\leqslant\cdots$.
4. You probably have alread realized this but $\displaystyle 1 + \cos \theta + \cos 2 \theta + \cos 3 \theta + ... + \cos n \theta= \frac{1}{2} + \frac{1}{2} \sum_{k=-n}^{n} e^{ik \theta}$