Nesbitt's inequality is a famous inequality with many unique solutions. It states that
Below is a list of proofs of Nesbitt's inequality. Feel free to add your own proofs.
Prove Nesbitt's Inequality
Proof 1:
Clearing denominators and full expansion gives that it is equivalent to
2a3+2b3+2c3≥a2b+a2c+b2a+b2c+c2a+c2b.
However, by AM-GM
3a3+a3+b3≥a2b,
so summing the inequality symmetrically gives the desired inequality.
□
Proof 2:
We have
b+ca+c+ab+a+bc=ab+aca2+bc+bab2+ca+cbc2≥2(ab+bc+ca)(a+b+c)2.
By Titu's lemma, it remains to prove
2(ab+bc+ca)(a+b+c)2≥23 or (a+b+c)2≥3(ab+bc+ca),
which is true after full expansion and the use of a2+b2+c2≥ab+bc+ca.
□
Proof 3:
Without losing generality we assume that a≥b≥c, which implies b+c1≥a+c1≥a+b1.
Now, applying Chebyshev's inequality, we have
3(b+ca+a+cb+a+bc)≥(a+b+c)(b+c1+a+c1+a+b1).
Applying Titu's lemma, we get a+b1+b+c1+a+c1≥2(a+b+c)9, so
b+ca+a+cb+a+bc≥23. □
Proof 4:
Let S denote the left side of the inequality, then we can transform S as follows:
LHS=S=b+ca+c+ab+a+bc=1+b+ca+1+c+ab+1+a+bc−3=b+ca+b+c+c+aa+b+c+a+ba+b+c−3=21[(a+b)+(b+c)+(c+a)](b+c1+c+a1+a+b1)−3.
Now, let x=a+b,y=b+c,z=c+a, then
2S⇒S=(x+y+z)(y1+z1+x1)−6=≥2yx+xy+≥2zx+xz+≥2zy+yz−3≥3=b+ca+a+cb+a+bc≥23. □