It is normal quick sort in which we are selecting the middle element as the pivot so the best case will be that it is divided into two equal parts giving the time complexity of O(nlogn).
In the worst case the it will take O(n^2) as the middle element selected can divide the array into two parts of 0 element at one side and n-1 at other.
now in the question it is asking for the lower bound in the best case.
the time complexity in best case is O(nlogn)
now tightest lower bound means the least possible time in which the problem can be solved.
in this case tightest lower bound is O(nlogn).