I would like to minimize the following function:
$$ \epsilon = \frac{1 - 1/n_0 \log(q_jq_i)}{(1-1/n_0\log(q_i))(1-1/n_0\log(q_j))} $$
where the two variables $q_i$ and $q_j$ have the following constraints, that are not independent of each other:
$$ \exp(\frac{b_k n_0(b_l-1)}{b_k(b_l-1)-b_l})< q_j < e^{n_0(1-b_l)} $$ $$ e^{n_0} q_j^{\frac{1}{b_l-1}} < q_i < e^{n_0(1-b_k)}q_j^{b_k-1} $$
All variables are real valued with $n_0 > 0$ and $1> b_k, b_l >0$. For simplicity it can also be assumed that $b_k = b_l = b$.
I would like to have an expression for $\epsilon_{min}(b_l,b_k,n_0)$ or in the simpler case $\epsilon_{min}(b,n_0)$.
The problem is, that the larger $q_j$ gets, the smaller is the maximal possible $q_i$, so there should be a trade of with an optimal point.
I would be thankful for any suggestions in how to approach this problem.