A lower bound for the on-line preemptive machine scheduling with lp norm

We consider the on-line version of the preemptive scheduling problem that minimizes the machine completion time vector in the l(p) norm (a direct extension of the l(infinity) norm: the makespan) on m parallel identical machines. We present a lower bound on the competitive ratio of any randomized on-line algorithm with respect to the general l(p) norm. This lower bound amounts to calculating a (non-convex) mathematical program and generalizes the existing result on makespan. While similar technique has been utilized to provide the best possible lower bound for makespan, the proposed lower bound failed to achieve the best possible lower bound for general l(p) norm (though very close), and hence revealing intricate and essential difference between the general l(p) norm and the makespan.