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 $\displaystyle \left|\sum_{i=1}^n a_ib_i\right|^2\leqslant \sum_{i=1}^n|a_i|^2\sum_{i=1}^n|b_i|^2$ then $\displaystyle \left|\sum_{i=1}^{n+1}a_ib_i\right|^2=\left|\sum_{ i=1}^{n}a_ib_i+a_nb_n\right|^2\leqslant\cdots$.
Finish.