oc.optimization and control – Optimality of a simple solution of a linear fractional minimax problem

Consider the following linear fractional optimization problem
begin{align}
max_mathbf{x}&quad min_{n=1,ldots,N}frac{x_n}{alpha+sum_{m}beta_m^{(n)}x_m}\
text{subject to}&quadmathbf{A}mathbf{x}leq mathbf{1},\
&quad mathbf{x}geq 0,
end{align}

where $mathbf{x}$ is a vector of $N$ non-negative variables. In addition, $alpha$ and $beta_m^{(n)}$ are positive constants and all the entries of matrix $mathbf{A}_{Ttimes N}$ are positive. $mathbf{1}$ is the all-one vector.

A simple lower-bound for the above problem is to consider a vector $mathbf{x}$ with equal entries. In this case, we have the following feasible point
begin{align}
mathbf{x}=xmathbf{1},
end{align}

where $x=frac{1}{max_{t=1,ldots,T}mathbf{A}mathbf{1}(t)}$. We now that this solution is optimal if for all $m,n$ we have $beta_m^{(n)}=0$. Could we anlytically bound the goodness of this feasible point?