<s>[INST] Does the inequality $I(n^2) \leq 2 - \frac{5}{3q}$ improve $I(q^k) + I(n^2) < 3$, if $q^k n^2$ is an odd perfect number with special prime $q$? Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Denote the abundancy index of the positive integer $x$ by $I(x)=\sigma(x)/x$, where $\sigma(x)=\sigma_1(x)$ is the classical sum of divisors of $x$.
Here is my initial question:

Does the inequality $I(n^2) \leq 2 - \frac{5}{3q}$ lead to an improvement to the upper bound $I(q^k) + I(n^2) < 3$, if $q^k n^2$ is an odd perfect number with special prime $q$?

MOTIVATION
Here is a proof that $I(n^2) \leq 2 - \frac{5}{3q}$ holds in general.
Suppose to the contrary that $I(n^2) > 2 - \frac{5}{3q}$ is true.
Note that
$$I(n^2) = \frac{2}{I(q^k)} = \frac{2q^k (q - 1)}{q^{k+1} - 1} = 2 - 2\cdot\Bigg(\frac{q^k - 1}{q^{k+1}-1}\Bigg),$$
so that we have
$$I(n^2) > 2 - \frac{5}{3q} \iff 2 - 2\cdot\Bigg(\frac{q^k - 1}{q^{k+1}-1}\Bigg) > 2 - \frac{5}{3q} \iff \frac{5}{3q} > 2\cdot\Bigg(\frac{q^k - 1}{q^{k+1}-1}\Bigg) \iff 5q^{k+1} - 5 > 6q^{k+1} - 6q \iff 0 > q^{k+1} - 6q + 5,$$
which then implies that $k=1$.  (Otherwise, if $k > 1$, we have
$$0 > q^{k+1} - 6q + 5 \geq q^6 - 6q + 5,$$
since $k \equiv 1 \pmod 4$, contradicting $q \geq 5$.)  Now, since $k=1$, we get
$$0 > q^2 - 6q + 5 = (q - 1)(q - 5),$$
which implies that $1 < q < 5$.  This contradicts $q \geq 5$.  This concludes the proof.
Now, let $Q = 2 - \frac{5}{3q}$.
Since $I(q^k) < I(n^2)$, then we obtain
$$I(q^k) < I(n^2) \leq Q \iff (I(q^k) - Q)(I(n^2) - Q) \geq 0$$
$$\iff 2 + Q^2 = I(q^k)I(n^2) + Q^2 \geq Q(I(q^k) + I(n^2) \iff I(q^k) + I(n^2) \leq \frac{2}{Q} + Q.$$
But $\frac{2}{Q} + Q$ can be rewritten as
$$\dfrac{2}{Q} + Q = \dfrac{2}{2 - \dfrac{5}{3q}} + \Bigg(2 - \dfrac{5}{3q}\Bigg) = 3 - \Bigg(\dfrac{5}{3q} - \dfrac{5}{6q-5}\Bigg) = 3 - \dfrac{5(3q - 5)}{3q(6q - 5)} = \dfrac{54q^2 - 60q + 25}{3q(6q - 5)}.$$
Let
$$f(q) = \dfrac{54q^2 - 60q + 25}{3q(6q - 5)}.$$
Then the derivative
$$f'(q) = \dfrac{5}{3q^2} - \dfrac{30}{(6q - 5)^2}$$
is positive for $q \geq 5$.  This means that $f$ is an increasing function of $q$, which implies that
$$I(q^k) + I(n^2) \leq \dfrac{2}{Q} + Q < \lim_{q \rightarrow \infty}{f(q)} = 3.$$
FINAL QUESTION

Can we do better?  If that is not possible, can you explain why?

 [/INST] Too long to comment :
There is no such upper bound of the form $2−\dfrac{cq+d}{q^2+aq+b}$ better than $2−\dfrac{2}{q+1}$.
Proof :
Suppose that there is $(a,b,c,d)$ such that
$$2 - 2\cdot\dfrac{q^k - 1}{q^{k+1} - 1} \leqslant 2 - \dfrac{cq+d}{q^2 + aq + b} \leqslant 2 - \dfrac{2}{q+1}$$
Then,
$$\dfrac{2q^k - 2}{q^{k+1} - 1} \geqslant \dfrac{cq+d}{q^2 + aq + b} \geqslant \dfrac{2}{q+1}$$
is equivalent to
$$\underbrace{\bigg((2-c)q^2+(2a-d)q+2b\bigg)}_{A}q^k-2q^2+(c-2a)q+d-2b\geqslant 0\tag1$$
$$(c-2)q^2+(c+d-2a)q+d-2b\geqslant 0\tag2$$
Suppose that $2-c\lt 0$. Then, it follows from $(1)$ that $A$ will be negative for a large $q^k$ and so LHS of $(1)$ will be negative. This is a contradiction. So, $2-c\geqslant 0$.
Suppose that $c-2\lt 0$. Then, it follows from $(2)$ that LHS of $(2)$ will be negative for large $q$. This is a contradiction. So, $c-2\geqslant 0$.
Therefore, we have to have $c=2$ for which we have
$$\underbrace{\bigg((2a-d)q+2b\bigg)}_{A}q^k-2q^2+(2-2a)q+d-2b\geqslant 0\tag3$$
$$(2+d-2a)q+d-2b\geqslant 0\tag4$$
It is necessary that $(3)$ holds for $k=1$, so we have to have
$$(2a-d-2)q^2+(2b+2-2a)q+d-2b\geqslant 0\tag5$$
Suppose that $2+d-2a\lt 0$. Then, LHS of $(4)$ will be negative for a large $q$. This is a contradiction. So, $2+d-2a\geqslant 0$.
Suppose that $2a-d-2\lt 0$. Then, LHS of $(5)$ will be negative for a large $q$. This is a contradcition. So, $2a-d-2\geqslant 0$.
Therefore, we have to have $d=2a-2$ for which we have
$$a-1-b\geqslant 0\tag6$$
$$(b+1-a)q+a-1-b\geqslant 0\tag7$$
Suppose that $b+1-a\lt 0$. Then, LHS of $(7)$ will be negative for a large $q$. This is a contradiction. So, $b+1-a\geqslant 0$.
Therefore, from $(7)$, we have to have $b=a-1$.
Now, since $b=a-1,c=2$ and $d=2a-2$, we finally have
$$2 - \dfrac{cq+d}{q^2 + aq + b}=2 - \dfrac{2q+2a-2}{q^2 + aq + a-1}=2-\frac{2(q+a-1)}{(q+1)(q+a-1)}=2-\dfrac{2}{q+1}$$
This means that there is no such upper bound of the form $2−\dfrac{cq+d}{q^2+aq+b}$ better than $2−\dfrac{2}{q+1}$.$\quad\blacksquare$

Similarly, one can prove that there is no such upper bound of the form $2−\dfrac{dq^2+eq+f}{q^3+aq^2+bq+c}$ better than $2−\dfrac{2}{q+1}$.
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