The Schwarz Lemma and Boundary Fixed Points

Abstract

Abstract. In this note we study the Schwarz lemma and its various versions. We find a condition for a holomorphic map to have fixed points only on the boundary of the unit disc and compare its derivatives at fixed points to get some relations among them.

Applications of the Jack's lemma for the meromorphic functions at the boundary

“…under the assumption f (0) = 0 where f is an analytic function mapping the unit disc into itself and c is a boundary point to which f extends continuously and |f (c)| = 1. Many studies have been carried out for the boundary Schwarz Lemma in the last 15 years (see, [1], [3], [4], [7], [8], [14], [15], [16], and references therein). Some of them are about the estimates from below for the modulus of the derivative of the function at the boundary points which satisfy the condition |f (c)| = 1.…”

“…Also, M. Jeong [7] got some inequalities at a boundary point for a different form of analytic functions and showed the sharpness of these inequalities. Also, M. Jeong found a necessary and sufficient condition for an analytic map having fixed points only on the boundary of the unit disc and compared its derivatives at fixed points to get some relations among them [8]. We refer to [1] for a more detailed explanation of the Schwarz Lemma and its applications on the boundary of the unit disc.…”

A Sharp Carathéodory's Inequality on the Boundary

“…Also, M. Jeong [8] showed some inequalities at a boundary point for different form of holomorphic functions and found the condition for equality and in [9] a holomorphic self map defined on the closed unit disc with fixed points only on the boundary of the unit disc.…”

Some lower bound for holomorphic functions at the boundary

“…In the last 15 years, there have been tremendous studies on Schwarz lemma at the boundary (see, [1], [3], [4], [7], [8], [10], [11], [16], [17], [18], [20] and references therein). Some of them are about the below boundary of modulus of the functions derivation at the points (contact points) which satisfies | f (b)| = 1 condition of the boundary of the unit circle.…”