  We present some formulae related to the Chern-Ricci curvatures and scalar
curvatures of special Hermitian metrics. We prove that a compact locally
conformal K\"{a}hler manifold with constant nonpositive holomorphic sectional
curvature is K\"{a}hler. We also give examples of complete non-K\"{a}hler
metrics with pointwise negative constant but not globally constant holomorphic
sectional curvature, and complete non-K\"{a}hler metric with zero holomorphic
sectional curvature and nonvanishing curvature tensor.
