Before revision:  We complement this by showing this bound is essentially tight: no online algorithm  can achieve a competitive ratio of O(\log^{1-\epsilon} N) or O(\log^{1-\epsilon} T) for any constant \epsilon>0.
After revision:  We complement this by showing this bound is essentially tight: no online algorithm , even if provided with perfectly accurate predictions, can achieve a competitive ratio of O(\log^{1-\epsilon} N) or O(\log^{1-\epsilon} T) for any constant \epsilon>0.