# Prove that: $\sqrt{x+y} + \sqrt{y+z} + \sqrt{z+x} \leq \sqrt{6(x+y+z)}$

Prove that for nonegative $$x,y,z$$ we have: $$\sqrt{x+y} + \sqrt{y+z} + \sqrt{z+x} \leq \sqrt{6(x+y+z)}$$

I prove that using the tangent line method. We may assume that $$x+y+z=1$$, so you we have to prove $$\sqrt{1-x}+\sqrt{1-y}+\sqrt{1-z}\leq \sqrt{6}$$

A tangent on $$f(x)=\sqrt{1-x}$$ at $$x={1\over 3}$$ is $$y=-{\sqrt{6}\over 4}x+{5\sqrt{6}\over 12}$$

So we have, for all $$x\in[0,1]$$:

$$\sqrt{1-x} \leq -{\sqrt{6}\over 4}x+{5\sqrt{6}\over 12}$$ and we are done...

I wonder if there is elegant method avoiding calculus?

• It is enough to apply Cauchy-Schwarz. Commented Oct 3, 2018 at 19:51

Hint: This is just $$(a+b+c)^2\leq 3(a^2+b^2+c^2)\iff ab+bc+ca\leq a^2+b^2+c^2$$ which is proved easily enough. You can start by $$a\equiv\sqrt{x+y}$$, $$b\equiv\ldots$$, etc.

• I just don't understand what is that equvalence for? Commented Oct 3, 2018 at 19:56
• @greedoid Your inequality can be rewritten as those 2 equivalent inequalities. Commented Oct 3, 2018 at 19:57

We may assume that $$x+y+z=1$$, so we have to prove $$\sqrt{1-x}+\sqrt{1-y}+\sqrt{1-z}\leq \sqrt{6}$$

$$\frac{\sqrt{1-x}+\sqrt{1-y}+\sqrt{1-z}}{3} \le \sqrt{\frac{\left(\sqrt{1-x}\right)^2+\left(\sqrt{1-y}\right)^2+\left(\sqrt{1-z}\right)^2}{3}} = \sqrt{\frac{2}{3}}$$

How about this: $$(\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x})^2=2(x+y+z)+2\left[\sqrt{(x+y)(y+z)}+\sqrt{(y+z)(z+x)}+\sqrt{(z+x)(x+y)}\right]\leq 2(x+y+z)+(x+2y+z)+(x+y+2z)+(2x+y+z)=6(x+y+z)$$ where one uses the GM-AM inequality. Finally, use $$0\leq x\leq y\implies \sqrt{x}\leq\sqrt{y}$$.

Let $$x+y+z=3$$.
Thus, we need to prove that $$\sum_{cyc}\sqrt{3-x}\leq3\sqrt2$$ or $$\sum_{cyc}\left(\sqrt2-\sqrt{3-x}\right)\geq0$$ or $$\sum_{cyc}\frac{x-1}{\sqrt2+\sqrt{3-x}}\geq0$$ or $$\sum_{cyc}\left(\frac{x-1}{\sqrt2+\sqrt{3-x}}-\frac{x-1}{2\sqrt2}\right)\geq0$$ or $$\sum_{cyc}\frac{(x-1)^2}{\left(\sqrt2+\sqrt{3-x}\right)^2}\geq0.$$ Also, by Jensen we obtain: $$\sum_{cyc}\sqrt{x+y}\leq3\sqrt{\frac{\sum\limits_{cyc}(x+y)}{3}}=\sqrt{6(x+y+z)}.$$