Motivated by the recent work of Ni on k-hyperbolicity of Kähler mani-folds,we investigate k-hyperbolicity of Hermitian manifolds.We prove a new Schwarz-lemma-type estimate for holomorphic map between complete Hermitian manfolds.As a corollary,we show that a compact Hermitian manifold with negative first k-Ricci curvature is k-hyperbolic.
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