  On an almost Hermitian manifold, we have two Hermitian scalar curvatures with
respect to any canonical Hermitian connection defined by P. Gauduchon. Explicit
formulas of these two Hermitian scalar curvatures are obtained in terms of
Riemannian scalar curvature, norms of decompositions of covariant derivative of
the fundamental 2-form with respect to the Levi-Civita connection, and the
codifferential of the Lee form. Then we get some inequalities of various total
scalar curvatures and some characterization results of the K\"{a}hler metric,
balanced metric, locally conformally K\"{a}hler metric and the k-Gauduchon
metric. As corollaries, we show some results related to a problem given by
Lejmi-Upmeier \cite{LeU} and a conjecture given by
Angella-Otal-Ugarte-Villacampa \cite{AOUV}.
