On the inequality $I(q^k)+I(n^2) \leq \frac{3q^{2k} + 2q^k + 1}{q^k (q^k + 1)}$ where $q^k n^2$ is an odd perfect number
Solution 1:
I think that the answer to the question (1) is no. Let $f(k):=\dfrac{3q^{2k}+2q^k+1}{q^k(q^k+1)}$. It follows from $I(q^k)+I(n^2)\leqslant f(1)$ that $I(q^k)+I(n^2)\leqslant f(k)$ since $f′(k)$ is positive.
I don't know how to prove the inequality in (2). Let $g(k):=\dfrac{3q^{2k} -4q^k + 2}{q^k (q^k -1)}$. Then, since $g'(k)$ is positive, one gets $g(1)\leqslant g(k)$. So, it does not follow from $I(q^k)+I(n^2)\gt g(1)$ that $I(q^k)+I(n^2)\gt g(k)$. This does not mean that it is not possible to prove $I(q^k)+I(n^2)\gt g(k)$.