Let $\sigma(x)$ denote the sum of the divisors of $x$, and let $I(x) = \sigma(x)/x$ be the abundancy index of $x$. A number $y$ is said to be almost perfect if $\sigma(y) = 2y - 1$.
In a preprint titled A Criterion For Almost Perfect Numbers Using The Abundancy Index, Dagal and Dris show that $N$ is almost perfect if and only if $$\dfrac{2N}{N + 1} \leq I(N) < \dfrac{2N + 1}{N + 1}.$$
Here is my question:
Are these bounds for $I(N)$ (when $N$ is almost perfect) best-possible?
UPDATE (September 27 2016)
In an answer below the fold, I was able to obtain the improved upper bound $$I(N) < \dfrac{4N}{2N+1}.$$ Can we likewise improve on the lower bound?