Prove that : $\displaystyle \frac{\sqrt{a^2+b^2}}{a+b}+\sqrt{\frac{ab}{a^2+b^2}}\le \sqrt{2}$ for all positive reals $a$ and $b$.

Prove that :
$\displaystyle \frac{\sqrt{a^2+b^2}}{a+b}+\sqrt{\frac{ab}{a^2+b^2}}\le \sqrt{2}$ for all positive reals $a$ and $b$.

My solution:

Step 1:

Squaring both sides of the inequality:

$\displaystyle \frac{\sqrt{a^2+b^2}}{a+b}+\sqrt{\frac{ab}{a^2+b^2}}\le \sqrt{2}$

$\displaystyle \left(\frac{\sqrt{a^2+b^2}}{a+b}+\sqrt{\frac{ab}{a^2+b^2}}\right)^2\le (\sqrt{2})^2$

$\displaystyle \frac{a^2+b^2}{(a+b)^2}+2\left(\frac{\sqrt{a^2+b^2}}{a+b}\cdot \sqrt{\frac{ab}{a^2+b^2}}\right)+\frac{ab}{a^2+b^2}\le 2$

$\displaystyle \frac{a^2+b^2}{(a+b)^2}+\frac{2\sqrt{ab}}{a+b}+\frac{ab}{a^2+b^2}\le 2$

$\displaystyle \frac{a^2+b^2}{(a+b)^2}+\frac{ab}{a^2+b^2}+\frac{2\sqrt{ab}}{a+b}\le 2$(*)

Up to this point, if we can prove the above inequality (*) as correct, then we're done.

Step 2:

Observe that the last term on the LHS of the inequality (*) above is less than or equal to 1, since:

$\displaystyle a+b\ge 2\sqrt{ab}$

$\displaystyle \frac{2\sqrt{ab}}{a+b}\le 1$

It remains to show $\displaystyle \frac{a^2+b^2}{(a+b)^2}+\frac{ab}{a^2+b^2}-1\le 0$(**).

Step 3:

If we distribute the $-1$ as two negative one half and give it to the two terms on the LHS of (**), we see that we have:

$\displaystyle \frac{a^2+b^2}{(a+b)^2}+\frac{ab}{a^2+b^2}-1$

$\displaystyle =\frac{a^2+b^2}{(a+b)^2}-\frac{1}{2}+\frac{ab}{a^2+b^2}-\frac{1}{2}$

$\displaystyle =\frac{2a^2+2b^2-(a+b)^2}{2(a+b)^2}+\frac{2ab-a^2-b^2}{2(a^2+b^2)}$

$\displaystyle =\frac{2a^2+2b^2-a^2-2ab-b^2}{2(a+b)^2}-\frac{a^2-2ab+b^2}{2(a^2+b^2)}$

$\displaystyle =\frac{a^2-2ab+b^2}{2(a+b)^2}-\frac{a^2-2ab+b^2}{2(a^2+b^2)}$

$\displaystyle =\frac{(a-b)^2}{2(a+b)^2}-\frac{(a-b)^2}{2(a^2+b^2)}$

$\displaystyle =(a-b)^2\left(\frac{1}{2(a+b)^2}-\frac{1}{2(a^2+b^2)}\right)$

$\displaystyle =(a-b)^2\left(\frac{a^2+b^2-(a+b)^2}{2(a+b)^2(a^2+b^2)}\right)$

$\displaystyle =(a-b)^2\left(-\frac{2ab}{2(a+b)^2(a^2+b^2)}\right)$

$\displaystyle \le 0$ since both $a$ and $b$ are positive real numbers and the inequality follows.