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In this paper, we derive some $\partial\overline{\partial}$-Bochner formulasfor holomorphic maps between Hermitian manifolds. As applications, we provesome Schwarz lemma type estimates, rigidity and degeneracy theorems. Forinstance, we show that there is no non-constant holomorphic map from a comapctHermitian manifold with positive (resp. non-negative) $\ell$-second Riccicurvature to a Hermitian manifold with non-positive (resp. negative) realbisectional curvature. These theorems generalize the results \cite{Ni1,Ni2}proved recently by L. Ni on K\"{a}hler manifolds to Hermitian manifolds. Wealso derive an integral inequality for holomorphic map between Hermitianmanifolds.

The $\partial\overline{\partial}$-Bochner formulas for holomorphic mappings between Hermitian manifolds and their applications