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Inequality of a triangle

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

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

Solution

Applying Cauchy’s inequality to the vectors

    \[\mathbf{u} = \left ( \sqrt{\sin A} , \sqrt{\sin B} , \sqrt{\sin C} \right )\]

and

    \[\mathbf{v} = \left ( \sqrt{\frac{1}{a^3+b^6+c^6}} , \sqrt{\frac{1}{b^3+c^6+a^6}} , \sqrt{\frac{1}{c^3+a^6+b^6}} \right )\]

we get that

    \begin{align*} \left ( \sum \sqrt{\frac{\sin A}{a^3+b^6+c^6}} \right )^2 &\leq \left ( \sum \sin A \right ) \left ( \sum \frac{1}{a^3+b^6+c^6} \right ) \\ &\leq \frac{3\sqrt{3}}{2} \sum \frac{1}{a^3+b^6+c^6} \end{align*}

due to the well known fact

(1)   \begin{equation*} \sin A + \sin B + \sin C \leq \frac{3\sqrt{3}}{2} \end{equation*}

It suffices to prove that \displaystyle \sum \frac{1}{a^3+b^6+c^6} \leq 1. Let x, y be two positive real numbers. Then,

(2)   \begin{equation*} \left ( x-y \right )^2\left ( x^4+x^3y + x^2y^2 + xy^3+ y^4 \right )\geq 0\Leftrightarrow x^6+y^6 \geq xy \left ( x^4+y^4 \right ) \end{equation*}

On the other hand if xyz \geq 1 then x + y + z \geq 3\sqrt[3]{xyz} =3 and

(3)   \begin{equation*} x^4+y^4+z^4 \geq \frac{\left ( x+y+z \right )^4}{27}  \end{equation*}

Then it follows that

    \begin{align*} \sum \frac{1}{a^3+b^6+c^6} &\leq \sum \frac{1}{ab\left ( a^4+b^4 \right )+c^3} \\ &\leq \sum \frac{1}{\frac{a^4+b^4}{c}+ c^3}\\ &= \sum \frac{c}{a^4+b^4+c^4} \\ &=\frac{a+b+c}{a^4+b^4+c^4}\\ &\leq \frac{27}{\left ( a+b+c \right )^3}\\ &\leq 1 \end{align*}

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