I am interested in bounding from above the ratio between the maximum and minimum entries of a Perron vector. The only results that I found in the literature are from the classic masters (Ostrowski and others) and can be seen here:

The nicest bound for my case seems to be $\frac{\max{x_{i}}}{\min{x{i}}} \leq \frac {R-r+m_{2}}{m_{2}}$, where $R,r$ are the largest and smallest rows sums and $m_{2}$ is smallest off-diagonal entry.

However, it is difficult for me to use this bound since I do not have an easy explicit form for the matrix. I do have the value of the Perron root.

So, I'd like to know if there has been any progress on this problem. (I am aware of a later paper by de Oliveira and a few recent papers that deal with special cases, but they appear to be inapplicable here).