I hope, I did not miss your point, the following Theorem with proof for general positive matrices is given in
"Nonnegative matrices" by Henryk Minc, Wiley Interscience Series in discrete mathematics and optimization, 1988, section 2.3, p. 41 - 43
The references cited there can be found on page 46.
Theorem:
Let $A \ = \ (a_{ij})$ be a positive matrix, with maximal eigenvector $x = (x_1, \ldots, x_n)$ and let $\gamma \ = \ \max_{i,j} (x_i/x_j)$ . Then
$ \sqrt{R/r} \ \leq \ \gamma \ \leq \ \max_{j,s,t} (a_{sj}/a_{tj}) \, ,$
where $R$ and $r$ are the greatest and least row sums of $A$, respectively. The left inequality is an equality if and only if $R = r$. Equality holds on the right-hand side if and only if the pth row of $A$ is a multiple of the qth row, for some pair of indices $p$ and $q$ satisfying $a_{ph}/a_{qh} \ = \ \max_{j,s,t} (a_{sj}/a_{tj}) \, .$
I just copied it with minor notational changes and did no proofreading.