  We examine various direct and indirect constraints on the lepton-specific
two-Higgs doublet model and scrutinize the property of the Higgs bosons in the
allowed parameter space. These constraints come from the precision electroweak
data, the direct search for Higgs boson, the muon anomalous magnetic moment, as
well as some theoretical consistency requirements. We find that in the allowed
parameter space the CP-odd Higgs boson A is rather light (m_A < 30 GeV with 95%
possibility), which is composed dominantly by the leptonic Higgs and decays
dominantly into \tau^+ \tau^-; while the SM-like Higgs boson h (responsible
largely for electroweak symmetry breaking) decays dominantly in the mode h->A
A->4 \tau with a large decay width, which will make the Higgs discovery more
difficult at the LHC. Whereas, this scenario predicts a branching ratio
Br(Z->\tau^+ \tau^- A) ranging from 10^{-5} to 10^{-4}, which may be accessible
at the the GigaZ option of the ILC.
