Show that $\sum_{\text{cyc}} \frac{1}{b^2+c^2+5bc-a^2} \leq \frac{\sqrt3}{8S}$ for a triangle with sides $a$, $b$, $c$ and area $S$

Question

Let be $a$, $b$, $c$ sides of a triangle and $S$ his area. Prove that $$\sum_{\text{cyc}} \frac{1}{b^2+c^2+5bc-a^2} \leq \frac{\sqrt3}{8S}$$

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My idea: $b^2+c^2-a^2 = 2bc \cos A$, so the inequality is equivalent to: $$\sum_{\text{cyc}} \frac{\sin A}{2\cos A+5}\leq\frac{\sqrt{3}}{4}$$

Answer 1

Now, you can use Jensen because $$\left(\frac{\sin\alpha}{5+2\cos\alpha}\right)''=\frac{\sin\alpha(10\cos\alpha-17)}{(5+2\cos\alpha)^3}<0$$ for all $0<\alpha<\pi.$