Inegalitate ,,nice'' 1

[Marius Mainea]

Fie a,b,c nenegative, cu a+b+c=1. Demonstrati :

\( \frac{1}{1+a^2}+\frac{1}{1+b^2}+\frac{1}{1+c^2}\geq\frac{5}{2} \)

Cu dedicatie pentru Claudiu Mandrila

[Claudiu Mindrila]

Din inegalitatea \( AM-GM \) avem ca \( a^3+a \geq 2a^2 \Rightarrow -\frac{a^2}{a^2+1} \geq -\frac{a}{2} \Rightarrow 1-\frac{a^2}{a^2+1} \geq 1-\frac{a}{2} \Rightarrow \frac{1}{a^2+1}\geq 1- \frac{a}{2}. \)
Rezulta ca \( \sum \frac{1}{a^2+1} \geq 3-\frac{a+b+c}{2}=3-\frac{1}{2}=\frac{5}{2} \), c.c.t.d.