How prove $\sum\frac{1}{2(x+1)^2+1}\ge\frac{1}{3}$ let $x,y,z>0$ and such $xyz=1$ show that
$$\dfrac{1}{2(x+1)^2+1}+\dfrac{1}{2(y+1)^2+1}+\dfrac{1}{2(z+1)^2+1}\ge\dfrac{1}{3}$$
My try: I will find a value of the $k$ such
$$\dfrac{1}{2(x+1)^2+1}\ge\dfrac{1}{9}+k\ln{x}$$
note $\ln{x}+\ln{y}+\ln{z}=0$,so
$$\sum_{cyc}\dfrac{1}{2(x+1)^2+1}\ge\dfrac{1}{3}+k(\ln{x}+\ln{y}+\ln{z})=\dfrac{1}{3}$$
so let
$$f(x)=\dfrac{1}{2(x+1)^2+1}-k\ln{x}-\dfrac{1}{9}$$
$$\Longrightarrow f'(x)=\dfrac{-4x-4}{(2x^2+4x+3)^2}-\dfrac{k}{x}$$
let $f'(1)=0\Longrightarrow k=-\dfrac{8}{81}$
so
$$f'(x)=\dfrac{-4x-4}{(2x^2+4x+3)^2}+\dfrac{8}{81x}=\dfrac{4(x-1)(8x^3+40x^2+15x-18)}{81x(2x^2+4x+3)^2}$$
so note when $1>x>\dfrac{1}{2}$ then
$$f'(x)=\dfrac{4(x-1)(8x^3+40x^2+15x-18)}{81x(2x^2+4x+3)^2}<0$$
$x>1,f'(x)>0$
so
$$f(x)\ge f(1)=0$$
so if $x,y,z>\dfrac{1}{2}$ we have prove done.
But for other case,How prove it? Thank you
 A: $z=\dfrac{1}{xy}$, put in LHS and and clean the denominators, we have:
edit:
LHS-RHS=$ 9y^2x^4-8y^3x^3+2y^2x^3+9y^4x^2+2y^3x^2-9y^2x^2-8yx^2-8y^2x+2yx+9 \ge0 \iff $
$4y^2x^4-8yx^2+4\ge 0,\\4y^4x^2-8y^2x+4\ge0,\\5y^2x^4+5y^4x^2\ge 10x^3y^3,\\2y^2x^3+2y^3x^2\ge 4(xy)^{\frac{5}{2}} \iff\\ LHS \ge 2x^3y^3+4(xy)^{\frac{5}{2}}-9x^2y^2+2xy+1=2t^6+4t^5-9t^4+2t^2+1=(t-1)^2(2t^4+8t^3+5t^2+2t+1) \ge0, t=\sqrt{xy}$
the "=" will hold when $xy=1,x=y,y^2x=yx^2=1 \implies x=y=z=1$
A: let $$x=\dfrac{bc}{a^2},y=\dfrac{ca}{b^2},z=\dfrac{ab}{c^2}$$
then we only prove follow  inequality
$$\dfrac{a^4}{3a^4+2b^2c^2+4a^2bc}+\dfrac{b^4}{3b^4+2c^2a^2+4b^2ca}+\dfrac{c^4}{4c^4+2a^2b^2+4c^2ab}\ge\dfrac{1}{3}$$
By Cauchy-Schwarz inequality 
\begin{align*}
\sum_{cyc}\dfrac{a^4}{3a^4+2b^2c^2+4a^2bc}&\ge\dfrac{(a^2+b^2+c^2)^2}{3(a^4+b^4+c^4)+2(a^2b^2+b^2c^2+a^2c^2)+4abc(a+b+c)}\\
&\ge\dfrac{(a^2+b^2+c^2)^2}{3(a^4+b^4+c^4)+2(a^2b^2+b^2c^2+c^2a^2)+4(a^2b^2+b^2c^2+c^2a^2)}\\
&=\dfrac{(a^2+b^2+c^2)^2}{3(a^2+b^2+c^2)^2}=\dfrac{1}{3}
\end{align*}
A: Hint:
Use Tringnometric transformation such that 
x=sinu, y=secu and z = cotu
For any value of u you can find x,y,z >0 such that xyz = 1 by definition.
For a value of u = pi/6, x=.5, y = 1.154 and z = 1.732
Further, I computed LHS, for u = pi/4 and its value is .3365
Just to double check, I computed LHS for u = pi/3 and its value is .34552
In both sample cases, LHS is >= (1/3).
Thanks
Satish
