Sharpened Versions of the Schwarz Lemma

Abstract

We prove a version of the Schwarz Lemma in which the images of two points are Ž . known. Two classical results due to Dieudonne and Rogosinski are simplé corollaries.

Schwarz-Pick-type estimates for the hyperbolic derivative

“…(iii) In [6], the author compares Theorem 3.1 with Schwarz-Pick Lemma 1.2. Proposition 3.2 may be similarly compared with Dieudonné's lemma (e.g., [2,4]), which refines Schwarz-Pick Lemma 1.2. A perfect analog of Dieudonné's lemma would read …”

The Schwarz Lemma and Boundary Fixed Points

“…Note that the equality in the above corollary holds only for Möbius transformation mapping the upper half plane into itself. When images of the two points are known, Mercer [9] got a sharpened version of the Schwarz-Pick lemma.…”

“…The Schwarz lemma is a simple and highly influential result in the theory of holomorphic mappings. Until now, it bears abundant fruits (see [3], [9], [12], [14]). …”

Sharpened forms of a von Neumann inequality for $\rho$-contractions

“…Having in mind that the inequality (3.5) for n = 1 is just the interior Schwarz lemma for analytic functions on the unit disc which appears in K. Fan [12], P. Mercer [17] and R. Osserman [19], we can also consider (3.2) and (3.3) as being sharpened forms of the interior Schwarz inequality for ρ-contractions. Now, in the case n = 1 we obtain from Theorem 3.1 the following result.…”