Recover approximate monotonicity of induced norms

Let $A$ some square matrix with real entries.
Take any norm $|cdot|$ consistent with a vector norm.

Gelfand’s formula tells us that $rho(A) = lim_{n rightarrow infty} |A^n|^{1/n}$.

Moreover, from [1], for a sequence of $(n_i)_{i in mathbb{N}}$ such that $n_i$ is divisible by $n_{i-1}$, we also know that the sequence $|A^{n_i}|^{1/n_i}$ is monotone decreasing and converges towards $rho(A)$. I am interested in what happens when this divisibility property is not verified.

1. If the matrix has non-negative entries, it seems the general property holds: For integers $n$ and $m$ such that $m > n$, it is the case that $|A^m|^{1/m} leq |A^n|^{1/n}$.

2. If the matrix can have positive and negative entries, this more general observation does not seem to hold. I am trying to understand why it fails, how worse can the inequality become, and if it is possible to recover an inequality up to some function of $A$: $|A^m|^{1/m} leq f(A)cdot|A^n|^{1/n}$.

Any references to 1., or pointers for understanding 2. would be much appreciated.

[1] Yamamoto, Tetsuro. "On the extreme values of the roots of matrices." Journal of the Mathematical Society of Japan 19.2 (1967): 173-178.