$1/(51 +u^2) + 1/(51 +v^2) + 1/(51 +w^2)$ inequality

Question

I need to prove that $$1/(51 +u^2) + 1/(51 +v^2) + 1/(51 +w^2) \leq 1/20$$ given $u + v + w = 9$ and $u,v,w$ positive reals.

Using AM-HM inequality with $(51 +u^2, 51 +v^2, 51 +w^2)$ I arrive at

$(51 +u^2 + 51 +v^2 + 51 +w^2)/3 \geq 3/((1/(51 +u^2) +1/(51 +v^2) 1/(51 +w^2))$, $$or,$$

$1/(51 +u^2) +1/(51 +v^2)+1/(51 +w^2) \leq 9/(51 +u^2 + 51 +v^2 + 51 +w^2)$

In order to take the next step, I need to estimate the RHS with an upper bound. Here is where I face the problem. With $(u+v+w)^2 \leq 3(u^2 + v^2 + w^2)$ from C-S inequality and $u+v+w =9$ we get $27 \leq (u^2 + v^2 + w^2)$ which does not provide an upper bound. I tried several other inequalities, but each lead to a bound on the "wrong" side. Am I at all looking in the right direction?

Answer 1

You wrote the following:

"Using AM-HM inequality with $(51 +u^2, 51 +v^2, 51 +w^2)$ I arrive at

$(51 +u^2 + 51 +v^2 + 51 +w^2)/3 \geq 3/((1/(51 +u^2) +1/(51 +v^2) 1/(51 +w^2))$, $$or,$$

$1/(51 +u^2) +1/(51 +v^2)+1/(51 +w^2) \leq 9/(51 +u^2 + 51 +v^2 + 51 +w^2)$"

See please better, the last line should be:

$1/(51 +u^2) +1/(51 +v^2)+1/(51 +w^2) \geq 9/(51 +u^2 + 51 +v^2 + 51 +w^2)$

You proved that $$\sum_{cyc}\frac{1}{51+u^2}\geq\frac{9}{\sum\limits_{cyc}(51+u^2)},$$ which does not help for the proof of your inequality.

By the way, the Tangent Line method helps.

Indeed, we need to prove that $$\frac{1}{20}-\sum_{cyc}\frac{1}{51+u^2}\geq0$$ or $$\sum_{cyc}\left(\frac{1}{60}-\frac{1}{51+u^2}\right)\geq0$$ or $$\sum_{cyc}\frac{(u-3)(u+3)}{51+u^2}\geq0$$ or $$\sum_{cyc}\left(\frac{(u-3)(u+3)}{51+u^2}-\frac{u-3}{10}\right)\geq0$$ or $$\sum_{cyc}\frac{(u-3)^2(7-u)}{51+u^2}\geq0,$$ which is true for $\max\{u,v,w\}\leq7.$

Let $u>7$.

Thus, $$\sum_{cyc}\frac{1}{51+u^2}<\frac{1}{51+7^2}+\frac{2}{51}<\frac{1}{20}$$ and we are done!

Answer 2

$27 \leq (u^2+v^2+w^2)$ is exactly what you want. You want an upper bound on $9/(153+u^2+v^2+w^2)$, so you need a lower bound on the denominator, which you just found.

Answer 3

You're doing well. Note that you have $u^2+v^2+w^2$ in the denominator, and $1/x$ is a decreasing function. That means that $$ \left( u^2+v^2+w^2 \ge \frac{(u+v+w)^2}{3} \right) \Rightarrow \left( \frac{9}{153+u^2+v^2+w^2} \le \frac{9}{153+\frac{(u+v+w)^2}{3}} \right)$$

EDIT: The answer is not correct, because the inequality $$\frac{1}{51+u^2} +\frac{1}{51+v^2}+ \frac{1}{51+w^2} \le \frac{9}{153+u^2+v^2+w^2} $$ is not correct. The correct one has $\ge$ sign, which doesn't help.