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Inequality in a triangle

Posted: Wed Sep 25, 2019 2:49 pm
by Riemann
Let $ABC$ be a triangle and denote $a, b, c$ the lengths of the sides $BC , CA$ and $AB$ respectively. If $abc \geq 1$ then prove that

$$\sqrt{\frac{\sin A}{a^3+b^6+c^6}} + \sqrt{\frac{\sin B}{b^3+c^6+a^6}} + \sqrt{\frac{\sin C}{c^3 + a^6+b^6}} \leq \sqrt[4]{\frac{27}{4}}$$