A note on Schwarz lemma for the modulus of holomorphic mappings on dd

In this note, we consider a holomorphic mapping f   from the unit disk DD in ℂ to p-ball Bp={z ∈ ℂn : ∑i=1n|zi|p < 1 }, 1 < p < +∞ It is proved that for such f, |∇||f||(z)|≤1-||f(z)||21-|z|2, z ∈ D.The extremal problem is also discussed when p is an even number. This result extends some related results on Schwarz lemma.