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