Show that $\frac 1 {\sqrt{x+y}}+\frac 1 {\sqrt{y+z}}+\frac 1 {\sqrt{z+x}}\geq2+\frac 1 {\sqrt2}$. 
Given $x,y,z\geq0$ and $xy+yz+zx=1$. Show that $\displaystyle\frac 1 {\sqrt{x+y}}+\frac 1 {\sqrt{y+z}}+\frac 1 {\sqrt{z+x}}\geq2+\frac 1 {\sqrt2}$.

I've tried many things but all failed. The only thing I know is that the equality holds when $x=y=1, z=0$. 
Please help. Thank you.
 A: Without loss of generality, we assume that $x\ge y\ge z$.

Lemma 1:
  $$\dfrac{1}{\sqrt{y^2+1}}+\dfrac{1}{\sqrt{z^2+1}}\ge 1+\dfrac{1}{\sqrt{(y+z)^2+1}}$$

Proof: this inequality is equivalent to
$$\dfrac{1}{y^2+1}+\dfrac{1}{z^2+1}+\dfrac{2}{\sqrt{(y^2+1)(z^2+1)}}\ge 1+\dfrac{1}{(y+z)^2+1}+\dfrac{2}{\sqrt{(y+z)^2+1}}.$$
Notice that
$$(y+z)^2+1-(y^2+1)(z^2+1)=yz(2-yz)\ge 0,$$
hence
$$\dfrac{2}{\sqrt{(y^2+1)(z^2+1)}}\ge\dfrac{2}{\sqrt{(y+z)^2+1}}.$$
Therefore, it suffices to prove that
$$\dfrac{1}{y^2+1}+\dfrac{1}{z^2+1}\ge1+\dfrac{1}{(y+z)^2+1}.$$
And this is equivalent to
$$\dfrac{yz[2-2yz-yz(y+z)^2]}{(y^2+1)(z^2+1)[(y+z)^2+1]}\ge 0.$$
Above is true because
$$2-2yz-yz(y+z)^2=2x(y+z)-yz(y+z)^2=(y+z)[2x-yz(y+z)]\ge (y+z)[2x-x^2(y+z)]=x(y+z)(2-xy-xz)\ge 0.$$


Lemma 2:
  $$\dfrac{1}{x+y}+\dfrac{1}{\sqrt{x+z}}\ge\sqrt{y+z}+\sqrt{\dfrac{y+z}{(y+z)^2+1}}.$$

Proof:
\begin{align}\dfrac{1}{x+y}+\dfrac{1}{\sqrt{x+z}}&=\sqrt{y+z}\left(\dfrac{1}{\sqrt{(y+z)(x+y)}}+\dfrac{1}{\sqrt{(z+x)(z+y)}}\right)\\
&=\sqrt{y+z}\left(\dfrac{1}{\sqrt{1+y^2}}+\dfrac{1}{\sqrt{1+z^2}}\right)
\end{align}
and apply Lemma 1, then done!

Thus the original inequality is equivalent to
$$\dfrac{1}{\sqrt{y+z}}+\sqrt{y+z}+\sqrt{\dfrac{y+z}{(y+z)^2+1}}\ge 2+\dfrac{1}{\sqrt{2}}.$$
Letting $$t=\dfrac{1}{\sqrt{y+z}}+\sqrt{y+z}\ge 2,$$ the inequality becomes
$$t+\dfrac{1}{\sqrt{t^2-2}}\ge 2+\dfrac{1}{\sqrt{2}}.$$
This can be shown by the following: 
\begin{align}
&t+\dfrac{1}{\sqrt{t^2-2}}-2-\dfrac{1}{\sqrt{2}}
\\
=&t+\dfrac{2\sqrt{2}}{2\sqrt{2}\cdot\sqrt{t^2-2}}-2-\dfrac{1}{\sqrt{2}}
\\
\ge& t+\dfrac{2\sqrt{2}}{2+t^2-2}-2-\dfrac{1}{\sqrt{2}}
\\
=&t+\dfrac{2\sqrt{2}}{t^2}-2-\dfrac{1}{\sqrt{2}}
\\
=&\dfrac{(t-2)(\sqrt{2}t^2-t-2)}{\sqrt{2}t^2}\ge 0.
\end{align}
A: $z=\dfrac{1-xy}{x+y} \ge 0 \to xy\le 1$
LHS$=\sqrt{\dfrac{1}{x+y}}+\sqrt{\dfrac{x+y}{1+x^2}}+\sqrt{\dfrac{x+y}{1+y^2}}=f(x,y)$
$f'_{x}=\dfrac{1-x^2-2xy}{(x^2+1) \sqrt{x^2+1} \sqrt{x+y}}+\dfrac{1}{\sqrt{(x+y)(y^2+1)}}-\dfrac{1}{(x+y)\sqrt{x+y}}=0$......<1>
$f'_{y}=\dfrac{1-y^2-2xy}{(y^2+1) \sqrt{y^2+1} \sqrt{x+y}}+\dfrac{1}{\sqrt{(x+y)(x^2+1)}}-\dfrac{1}{(x+y)\sqrt{x+y}}=0$......<2>
<1>$*(1+y^2)$-<2>$*(1+x^2)$:
$(y^2-x^2)\left(2(x+y)(1-xy)+\dfrac{(1+x^2)(1+y^2)}{\sqrt{1+x^2}{1+y^2}}\right)=0$
so we have $x=y$ only. put in <1>, we get: $4x-4x^3=(x^2+1)^{\frac{3}{2}}$,ie:
$15x^6-35x^4+13x^2-1=0 \to (3x^2-1)(5x^4-10x^2+1)=0$, we get two roots:
$x_1=\dfrac{1}{\sqrt{3}}, x_2=\sqrt{1-\dfrac{2}{\sqrt{5}}}$
$f(x_1)=2.79,f(x_2)=2.77$,but we note $f(x,y)$ when $x ,y \to \infty, f(x,y) \to 0$,so we have to check the  bound, which is $xy=1$. 
$f(x,y)=\sqrt{x}+\dfrac{1}{\sqrt{x}}+\dfrac{1}{\sqrt{\dfrac{1}{x}+x}}$
let $m=\sqrt{x}+\dfrac{1}{\sqrt{x}} \ge2$
$f(m)=m+\dfrac{1}{\sqrt{m^2-2}}$, when $m\ge 2$, $f(m)$ is mono increasing, so the minimum will be got when $m=2 \to x=y=1, f(1)=2+\dfrac{1}{\sqrt{2}}< 2.77$, so the  minimum is $2+\dfrac{1}{\sqrt{2}}$.
To prove $f(m)=m+\dfrac{1}{\sqrt{m^2-2}}$ is mono increasing function, 
$f'(m)=1-\dfrac{m}{(m^2-2)^{3/2}}=1-\dfrac{1}{\sqrt{m^2-2}\left(m-\dfrac{2}{m}\right)}>0$ when $m\ge 2$.
