Is anything like $\dfrac n{\phi(n)}<\dfrac{\sigma(n)}n<e^\gamma\log\log n$ known/conjectured for the generalizations of these functions?

Let $n=p_1^{a_1}\cdots p_t^{a_t}$ be the canonical prime factorization of $n$. Combining these bounds we get the well-known inequality, $\dfrac{\phi(n)\sigma(n)}{n^2}<1$, but from the definitions

$$J_k(n)=\prod_{i=1}^t\left(p_i^{ka_i}-p_i^{k(a-1)}\right)$$

and

$$\sigma_k(n)=\prod_{i=1}^t\dfrac{p_i^{k(a_i+1)}-1}{p_i^k-1}$$

it's clear that

$$\dfrac{J_k(n)\sigma_k(n)}{n^{2k}}=\prod_{i=1}^t\left(1-p_i^{-k(a_i+1)}\right)<1$$

Are any other good pairs of upper bounds for $J_k(n)$ and $\sigma_k(n)$ known for $k\ne1$ of a form similar to those for $k=1$?

Edit: I figure we should have something like $\dfrac{n^k}{J_k(n)}<\dfrac{\sigma_k(n)}{n^k}<B_k(n)$, where $B_k(n)$ is the upper bound in question. As in the case $k=1$, it may be known for the totient function, but not for the divisor function.