We have $n$ characteristic functions with: \begin{equation} \label{eq:character} u^2 - ((1+\beta)(1-\eta \lambda_i))u + \beta(1-\eta \lambda_i)= 0, \end{equation} where $0 \leq m \leq \lambda_i \leq L$ ; $i\in [n]$ ; $\beta , \eta \geq 0$.
Now we need to find the optimal parameters $\eta$ and $\beta$ to meet: \begin{eqnarray} \mathop{\min}_{\eta, \beta}\mathop{\max}_{i \in [n]}\{|u_i^1|, |u_i^2|\} < 1 \end{eqnarray} where $u_i^1$ and $u_i^2$ are the roots of the above characteristic function based on the $\lambda_i$