Doua inegalitati cu numere complexe

[Razvan Balan]

1. Demonstrati ca \( 0 \leq (z_1+z_2+...+z_n)(\frac{1}{z_1}+\frac{1}{z_2}+...+\frac{1}{z_n}) \leq n^2 \), unde \( |z_1|=|z_2|=...=|z_n|=1 \)

Gh. Andrei

2. \( |z_0-z_1|^2+|z_0-z_2|^2+...+|z_0-z_n|^2 \leq |z-z_1|^2+ |z-z_2|^2+...+|z-z_n|^2, \) unde \( z_0=\frac{z_1+z_2+...+z_n}{2}, z,z_1,z_2,...,z_n \in\mathbb{C}, n \geq 2 \).

Gh. Andrei

[Razvan Balan]

1. \( (\sum_{j=1}^n z_j)(\sum_{j=1}^n \frac{1}{z_j})=(\sum_{j=1}^n z_j)(\sum_{j=1}^n \bar z_j )= | \sum_{j=1}^n z_j |^2 \) si atunci inegalitatea devine \( 0\leq|\sum_{j=1}^n z_j |^2 \leq n^2=(\sum_{j=1}^n |z_j|)^2 \) si se dovedeste a fi adevarata.