In a related question, I had asked about the norm induced by an inverse Wishart matrix. I am interested in generalizing that result somewhat. Let $A\sim\mathcal{W}_p\left(I,n\right)$, a Wishart matrix with scale matrix $I$, the identity, and $n$ degrees of freedom. Let $l$ be some fixed $p$-dimensional vector. Consider $$h = \frac{l^{\top}l}{l^{\top} \left(A^{-1}\right)^{m} l},$$ where $m$ is some integer. When $m=1$, $h$ is evidently distributed as a Chi-square random variable (with $n-p+1$ degrees of freedom, I believe).

Are there known results on the distribution of $h$ when $m=2$? Other values?

In a related question, I had asked about the norm induced by an inverse Wishart matrix. I am interested in generalizing that result somewhat. Let $A\sim\mathcal{W}_p\left(I,n\right)$, a Wishart matrix with scale matrix $I$, the identity, and $n$ degrees of freedom. Let $l$ be some fixed $p$-dimensional vector. Consider $$h = \frac{l^{\top}l}{l^{\top} \left(A^{-1}\right)^{m} l},$$ where $m$ is some integer. When $m=1$, $h$ is evidently distributed as a Chi-square random variable (with $n-p+1$ degrees of freedom, I believe).

Are there known results on the distribution of $h$ when $m=2$? Other values?