Bounding the row sum arithmetic mean by Perron roots of row-permuted matrices

$R_+^{n\times n}$ denotes the set of $n\times n$ non-negative matrices. For $A\in R_+^{n\times let $\Omega(A)$ be all matrices that can formed by permuting elements within each row $A$. Formally: $$\Omega(A)=\{B\in n}: \forall i\;\exists\text{ a permutation }\phi_i\; \text{s.t.}\ b_{i,j}=a_{i,\phi_i(j)}\;\forall j\}.$$ $B\in\Omega(A)$ $\rho(B)$ denote spectral radius or largest non negative eigenvalue $B$. We show arithmetic mean sums $A$ is bounded maximum and minimum in Formally, we are showing $$\min_{B\in\Omega(A)}\rho(B)\leq \frac{1}{n}\sum_{i=1}^n\sum_{j=1}^n a_{i,j}\leq \max_{B\in\Omega(A)}\rho(B).$$ positive also obtain necessary sufficient conditions for one these inequalities (or, equivalently, both them) to become an equality. give criteria which irreducible matrix $C$ should satisfy have $\rho(C)=\min_{B\in\Omega(A)} \rho(B)$ $\rho(C)=\max_{B\in\Omega(A)} \rho(B)$. These used derive algorithms finding such when entries .