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We prove two results related to the Schwarz lemma in complex geometry. First,we show that if the inequality in the Schwarz lemmata of Yau, Royden andTosatti becomes equality at one point, then the equality holds on the wholemanifold. In particular, the holomorphic map is totally geodesic and hasconstant rank. In the second part, we study the holomorphic sectional curvatureon an almost Hermitian manifold and establish a Schwarz lemma in terms ofholomorphic sectional curvatures in almost Hermitian setting.

Schwarz lemma: the case of equality and an extension