Inequality. $\sqrt{\frac{11a}{5a+6b}}+\sqrt{\frac{11b}{5b+6c}}+\sqrt{\frac{11c}{5c+6a}} \leq 3$ 
Let $a,b,c$ be positive numbers . Prove the following inequality:
$$\sqrt{\frac{11a}{5a+6b}}+\sqrt{\frac{11b}{5b+6c}}+\sqrt{\frac{11c}{5c+6a}} \leq 3.$$

What I tried:
I used Cauchy-Schwarz in the following form
$\sqrt{Ax}+\sqrt{By}+\sqrt{Cz} \leq \sqrt{(a+b+c)(x+y+z)}$ for:
$$A=11a, \quad{} B=11b, \quad{} C=11c$$
and
$$x=\frac{1}{5a+6b}, \quad{} y=\frac{1}{5b+6c}, \quad{} z=\frac{1}{5c+6a}$$ but still nothing. Thanks for your help :)
I tried something else:
$$\large\frac{\sqrt{\frac{11a}{5a+6b}}+\sqrt{\frac{11b}{5b+6c}}+\sqrt{\frac{11c}{5c+6a}}}{3}  \leq \sqrt{\frac{\frac{11a}{5a+6b}+\frac{11b}{5b+6c}+\frac{11c}{5c+6a}}{3}}$$ and what we have to prove become:
$$\large\sqrt{\frac{\frac{11a}{5a+6b}+\frac{11b}{5b+6c}+\frac{11c}{5c+6a}}{3}} \leq 1 \Leftrightarrow \sqrt{\frac{11a}{5a+6b}+\frac{11b}{5b+6c}+\frac{11c}{5c+6a}} \leq \sqrt{3}$$
Another attempt
$$\large\sqrt{\frac{1}{xy}} \leq \frac{\frac{1}{x}+\frac{1}{y}}{2}=\frac{x+y}{2xy}$$
and $y=1$ and $\displaystyle x=\frac{5a+6b}{11a}$. So:
$$\large\sqrt{\frac{11a}{5a+6b}} \leq \frac{\frac{5a+6b}{11a}+1}{2 \cdot \frac{5a+6b}{11a}}=\frac{8a+3b}{5a+6b}.$$ Now we have to prove:
$$\sum_{cyc}{\frac{8a+3b}{5a+6b}} \leq 3.$$
But still nothing .
 A: We can prove the more general inequality for certain values of the ratio between b and a:

$$\sqrt{\frac{(a+b)x}{ax+by}}+\sqrt{\frac{(a+b)y}{ay+bz}}+\sqrt{\frac{(a+b)z}{az+bx}} \leq 3$$
  if  0.8152 < b/a < 1.2267

Re-working the LHS and applying Cauchy-Schwarz we get:
$$(\sqrt{az + bx}\sqrt{\frac{(a + b) x}{(a z + b x) (a x + b y)}} 
+ \sqrt{ax + by}\sqrt{\frac{(a + b) y}{(a x + b y) (a y + b z)}}
+ \sqrt{ay + bz}\sqrt{\frac{(a + b) z}{(ay + bz) (a z + b x)}})^2 
\leq 
(a y + b z + a z + b x + a x + b y)(\frac{(a + b) x}{(a z + b x) (a x + b y)}+\frac{(a + b) y}{(a x + b y) (a y + b z)}+\frac{(a + b) z}{(ay + bz) (a z + b x)})
=\frac{(a + b)^3 (x + y + z) (y z + x y + xz)}{(a x + b y) (b x + 
   a z) (a y + b z)}
$$
We are now looking for an upper bound of this last expression. Denote this upper bound by k and consider the expression:
$$G=(a + b)^3 (x + y + z) (y z + x y + zx) - k (a x + b y) (b x + a z) (a y + b z)$$
This needs to be always negative if k is an upper bound.
Simplifying it turns out that we need a value of k such that:
$$G=((a+b)^3-a^2 b k)(y^2 z+x z^2+x^2 y)+((a+b)^3-a b^2 k)(xy^2+yz^2+zx^2)+(3 (a+b)^3-a^3 k-b^3 k)xyz<=0$$
Applying AM/GM we get:
$$(a^2 b k-(a+b)^3)(y^2 z+x z^2+x^2 y)+(a b^2 k-(a+b)^3)(xy^2+yz^2+zx^2)>=3((a^2 b k-(a+b)^3)+(a b^2 k-(a+b)^3))xyz$$
In order for this to hold we need to assume that (A1) $$a^2 b k-(a+b)^3>=0$$ and (A2) $$a b^2 k-(a+b)^3>=0$$
So now we can choose k so that the coefficients before xyz in the last two expressions are equal. In other words we need k such that:
$$3(a + b)^3 - a^3 k - b^3 k =
 3 ((a^2 b k - (a + b)^3) + (a b^2 k - (a + b)^3))$$
It is easy to see that k=9 we completes the proof if our assumptions (A1) and (A2) hold.
They hold for a small range of b/a ratios - approximately 0.8152 < b/a < 1.2267.
In order for A1 and A2 to hold for k=9 we can rewrite them in the form $$\frac{(a + b)^3}{a^2 b}<=9 \quad and \quad \frac{(a + b)^3}{a b^2}<=9$$
Letting x=b/a this means that we need the positive values of x for which both:$$
1/x + 3 x + x^2 - 6<=0 \quad and \quad 1/x^2 + 3/x + x - 6<=0$$
Solving the resulting cubics we get:$$
2 - \sqrt{3} Cos(\pi/18) + 3 Sin(\pi/18)\le\frac{b}{a}\le-1 + 3 Cos(\pi/9) - \sqrt{3} Sin(\pi/9)
$$ 
or numerically 0.8152 < b/a < 1.2267. This is the range of possible valus for b/a for which our inequality always holds.
You can also note that the product of the interval ends is equal to 1.
A: It's a bit ugly (actually more than a bit), but it works.
By AM-GM we get
$$
abc = \sqrt[3]{ab^2\cdot bc^2 \cdot ca^2} \leq \frac 1 3 (ab^2 + bc^2 + ca^2)
$$
and therefore
$$
9(5a + 6b)(5b + 6c)(5c + 6a) - 11^3(a + b + c)(ab + bc + ca) =\\ 289(ab^2 + bc^2 + ca^2) + 19(a^2b + b^2c + c^2a) - 924abc =\\ 289(ab^2 + bc^2 + ca^2 - 3abc) + 19(a^2b + b^2c + c^2a - 3abc) \geq 0
$$
From the previous inequality and applying Cauchy-Schwarz we arrive to
$$
\sum_{cyc}\sqrt{\frac {11(5c + 6a)} {(5a + 6b)(5b + 6c)(5c + 6a)} \cdot a(5b + 6c)} \leq\\
 \sqrt{\frac {11^2 (a + b+ c)} {(5a + 6b)(5b + 6c)(5c + 6a)} \sum_{cyc} a(5b + 6c) }  =\\
 \sqrt{\frac {11^3 (a + b+ c) (ab + bc + ca)} {(5a + 6b)(5b + 6c)(5c + 6a)}} \leq \sqrt 9 = 3
$$
