Inegalitate cu variabile >0 (OWN)

[Claudiu Mindrila]

Daca \( a,\ b,\ c>0 \), demonstrati ca \( \frac{a}{\sqrt{b^{2}+c^{2}+2}}+\frac{b}{\sqrt{c^{2}+a^{2}+2}}+\frac{c}{\sqrt{a^{2}+b^{2}+2}}\le\frac{a^{2}+b^{2}+c^{2}}{2\sqrt{abc}}
\)

(Claudiu Mindrila, R. M. T. 3/2009)

[Marius Mainea]

Notam \( a=x^4, b=y^4, c=z^4 \) si aplicam inegalitate mediilor

\( b^2+c^2+2=y^8+z^8+2\ge 2y^4z^4+2\ge2\cdot 2 y^2z^2 \)

Atunci \( LHS\le \frac{x^4}{2zx}+\frac{y^4}{2zx}+\frac{z^4}{2xy}\le \frac{x^8+y^8+z^8}{2x^2y^2z^2}=RHS \)