Inegalitate

[maxim bogdan]

Daca \( a,b,c,x,y,z,t \) sunt numere reale pozitive, demonstrati inegalitatea:

\( \frac{1}{ax+by+cz}+\frac{1}{ay+bz+ct}+\frac{1}{az+bt+cx}+\frac{1}{at+bx+cy}>\frac{8\sqrt{3}}{\sqrt{a^2+b^2+c^2}\cdot\sqrt{x^2+y^2+z^2+t^2}}. \)

D.M. Batinetu-Giurgiu

[Marius Mainea]

\( LHS\ge\frac{(1+1+1+1)^2}{\sum {ax+by+cz}}=\frac{16}{(a+b+c)(x+y+z+t)}\ge \frac{16}{\sqrt{3(a^2+b^2+c^2)}\sqrt{4(x^2+y^2+z^2+t^2)}}=\frac{RHS}{3} \)