KöMaL 2015 B4692

The sides of an acute-angled triangle are $a$ , $b$ , $c$ , the opposite angles are $\alpha$, $\beta$, $\gamma$, and the lengths of the corresponding altitudes are $m_a$, $m_b$, $m_c$, respectively. Prove that $\frac{m_a}{a} + \frac{m_b}{b} + \frac{m_c}{c} \ge 2\cos \alpha \cos\beta \cos \gamma \left(\frac{1}{\sin 2\alpha} + \frac{1}{\sin 2\beta} + \frac{1}{\sin 2\gamma}\right) + \sqrt{3}\,$.