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?