How to solve the following mixed recurrence relations:
$$T(n)= \max_{0<k<n} \left\{ \min \left\{ k, a T\left(\frac{n-k}{b}\right)\right\} \right\}$$
Here $a,b \ge 1$ are constants.
I could solve some special cases such as when $a=b=2$ then by guessing method, $T(n) = \Theta (\dfrac n {\lg n})$. However, I could not generalize for all values of $a,b$.