How to prove a generalised function is quasiconcave?
Note first that the function $r\mapsto A(r)^{\mu/p}$ is strictly increasing on $\mathbb{R}_+$. When you look at the minimum in the definition of quasi-concavity, you can therefore ignore this part and it suffices to show that the function
$$(x_1,x_2,\ldots,x_n)\mapsto a_1x_1^p+a_2x_2^p+...+a_nx_n^p$$
is quasi-concave. Actually, the function is even concave. The sum of concave functions is concave, so it suffices to show that the function
$$(x_1,x_2,\ldots,x_n)\mapsto a_ix_i^p$$
is concacave for each $i=1,\ldots,n$.