Valid Inequality for a Triangle!

$\large{\left( \dfrac{h_a^t + h_b^t + h_c^t}{3} \right)^{1/t} \leq \dfrac{\sqrt{3}}{2}\left( \dfrac{a^t + b^t + c^t}{3}\right)^{1/t}}$

Let $a,b,c$ be the sides of a triangle and $h_a, h_b, h_c$ respectively be the corresponding altitudes. If the maximum range of validity of the above inequality for $t$ be: $\quad \large{-\alpha < t < \alpha}$ ,

where $t \neq 0$ and $\alpha = \dfrac{\ln(A)}{\ln(B)}$ , and where $A,B \in \mathbb R$ , find the value of $\dfrac{A}{B}$ .