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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:

http://www.jstor.org/discover/10.2307/2949660?uid=3738232&uid=2129&uid=2&uid=70&uid=4&sid=47699037112127

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).

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Take two arbitrary positive column vectors $l$ and $r$ and consider the matrix $rl^t$ to see by scaling one of the vectors, that the Perron root alone will give no control of the Birkhoff norm of the Perron-Frobenius-vectors. I recently answered an essentially duplicate version of your question: <mathoverflow.net/questions/161145/…;, but you know that material yourself. I would be glad to hear about any recent results in this direction as I might need them myself, so I upvoted this one. –  thomashennecke Mar 29 at 12:50
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