  The exponential synchronization rate is addressed for Kuramoto oscillators in
the presence of a pacemaker. When natural frequencies are identical, we prove
that synchronization can be ensured even when the phases are not constrained in
an open half-circle, which improves the existing results in the literature. We
derive a lower bound on the exponential synchronization rate, which is proven
to be an increasing function of pacemaker strength, but may be an increasing or
decreasing function of local coupling strength. A similar conclusion is
obtained for phase locking when the natural frequencies are non-identical. An
approach to trapping phase differences in an arbitrary interval is also given,
which ensures synchronization in the sense that synchronization error can be
reduced to an arbitrary level.
