A Geometric Extension of Schwarz’s Lemma and Applications

Let f be a holomorphic function of the unit disc , preserving the origin. According to Schwarz’s Lemma, |f'(0)| ≤ 1, provided that . We prove that this bound still holds, assuming only that f() does not contain any closed rectilinear segment [0, eiϕ], ϕ ∊ [0, zπ], i.e., does not contain any entire radius of the closed unit disc. Furthermore, we apply this result to the hyperbolic density and give a covering theorem.