Proving inequality $\frac{a}{\sqrt{a+2b}}+\frac{b}{\sqrt{b+2c}}+\frac{c}{\sqrt{c+2a}}\lt \sqrt{\frac{3}{2}}$

This is too long for a comment but it's more of a suggestion than an answer. Without loss of generality let $a \leq b \leq c. $ Put $a = x,\ b = y,\ c = z = 1 - x - y$:

Let $$w(x,y) = \frac{x}{\sqrt{x+2y}} + \frac{y}{\sqrt{y+2(1-x-y)}} + \frac{1-x-y}{\sqrt{1-x-y+2x}} $$

The constraint $a + b + c = 1$ means that x can be no larger then 1/3, otherwise y must be greater than 1/3, and z is forced to be less than 1/3 contrary to assumption. So $0 < x \leq 1/3.$ If x is minimally (almost) $0$, y is at most $1/2$ otherwise z is less than y contrary to assumption. So $x\leq y \leq 1/2.$

So the problem is now:

Maximize w(x,y)

subject to

$0 < x\leq 1/3$

$x\leq y \leq 1/2$

If the maximum we find is less than $\sqrt{\frac{3}{2}}$ the original inequality is true.

We should verify (formally) the visual evidence of a 3D plot of the feasible region, which is that in $\{0 < x\leq 1/3, x\leq y \leq 1/2\}$ the function $w(x,y)$ has a local maximum on $y = 0^+$ (a last reminder that $0<x,y$) for suitable choice of x. Then to find that maximum we can let y equal $0$ and set the derivative $w_x(x,0) $ equal to zero. This (using a numerical routine) gives $x\approx 0.1547.$ A plot of $w(x, 0)$ shows this to be a maximum at about 1.179.

The value of $\sqrt{3/2}$ is about 1.22, so $w(x,y)\ll \sqrt{3/2}$ for this point.

Again, using the visual shortcut, there also appears to be a local maximum for $w(x,y)$ on $x=0$ for suitable choice of y. If we let $w_y(0,y)= 0$ we (again numerically) obtain a value of $y \approx 0.845,$ at which $w(0,y)$ appears to be a maximum, but this is outside the feasible region, and the maximum is attained in the feasible region at $w(0,1/2) = 1.115,$ which is less than 1.179 and not maximal for the region as a whole.

So this sketch of an argument, which omits some important formalities, suggests the inequality is true.

Edit: Khue noted a problem with this answer and suggested a simple fix. We cannot without generality assume $x < y < z,$ but can assume $x =\text{ min}(x,y,z),$ then the problem is minimize $w(x,y)$ subject to:

$0\leq x \leq 1/2$, and $x \leq y \leq 1.$

As Khue notes, this changes the feasible region. Again we can plot the feasible region and the max appears to occur along $x = 0, 0 \leq y \leq 1.$ As before finding $w_y(0,y)$ the max occurs at about $y = 0.845299$ and at that value $w = 1.17996.$