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.
Categories: math.DG math.CV