+0  
 
0
81
1
avatar

Prove that if a, b, and c are positive real numbers, then

sqrt(a^2 + ab + b^2) + sqrt(a^2 + ac + c^2) + sqrt(b^2 + bc + c^2) >= sqrt(3) (sqrt(ab) + sqrt(ac) + sqrt(bc))

When does equality occur?

 Mar 27, 2023
 #1
avatar+397 
+1

Consider the terms \(a^{2},ab,b^{2}.\).

Their arithmetic mean will be greater than or equal to their geometric mean, (equality when \(a=b),\)

so

\(\displaystyle \frac{a^{2}+ab+b^{2}}{3} \geq\sqrt[3]{a^{2}.ab.b^{2}}\\ a^{2}+ab+b^{2}\geq3ab \\\sqrt{a^{2}+ab+b^{2}}\geq\sqrt{3}\sqrt{ab}.\)

Repeat for the other two and add together all three.

Equality when a = b = c.

 Mar 27, 2023

2 Online Users

avatar