  The Schwarz lemma for holomorphic maps between Hermitian manifolds is
improved. New curvature constraints on the source and target manifolds are
introduced and shown to be weaker than the Ricci and real bisectional
curvature, respectively. The novel target curvature condition is intrinsic to
the Hermitian structure and is controlled by the holomorphic sectional
curvature if the metric is pluriclosed. This leads to significant improvements
on the Wu--Yau theorem. Further, it is shown that the Schwarz lemma is largely
invariant under a change of Hermitian connection, and the precise geometric
quantity that varies with the connection is determined. This enables us to
establish the Schwarz lemma for the Gauduchon connections.
