Inegalitati in C
Posted: Sat Nov 08, 2008 8:47 pm
Demonstrati ca \( |z_1+z_2| + |z_2+z_3| + |z_1+z_3| \le |z_1| + |z_2| + |z_3| + |z_1+z_2+z_3| \) ,pentru \( \forall z_1, z_2, z_3 \in \mathbb C \)
(\( |z_1z_2|+|z_2+z_3|+|z_1+z_3|)^2=|z_1+z_2|^2+|z_2+z_3|^2+|z_1+z_3|^2+2(|z_1+z_2| |z_2+z_3|+|z_1+z_2| |z_1+z_3|+|z_2+z_3| |z_1+z_3| (1) \)
\( |z_1+z_2| |z_2+z_3|=|z_1z_2+z_1z_3+z_2^2+z_2z_3|=|z_2(z_1+z_2+z_3)+z_1z_3)\le|z_2| |z_1+z_2+z_3|+|z_1||z_3| (2) \)
\( Analog<br>
|z_1+z_2| |z_1+z_3|\le|z_1| |z_1+z_2+z_3|+|z_2||z_3| (3) \)
\( |z_2+z_3||z_1+z_3|\le|z_3||z_1+z_2+z_3|+|z_1||z_2|(4) \)
\( (2)(3)(4) => \)
\( |z_1+z_2| |z_2+z_3|+z_1+z_2| |z_1+z_3|+|z_2+z_3||z_1+z_3|\le|z_1+z_2+z_3|(|z_1+z_2+z_3|+|z_1| |z_3|+|z_1||z_2|+|z_1| |z_3|) \)
\( (1)\le(o mai prelucrezi putin)............=(|z_1|+|z_2|+|z_3|+|z_1+z_2+z_3)^2 \) \( => \) ceea ce trebuia \( demonstrat \)
sper sa nu fi gresit in redactare
daca gasesti o alta metoda sa o scri