Note on Schwarz-Pick estimates for bounded and positive real part analytic functions

Abstract

Abstract. In this note we consider higher order derivatives of bounded analytic functions.

On higher order angular derivatives – an application of Faà di Bruno's formula

“…The proof follows the steps similar to the proof of Theorem 1.3 in [2]. First consider the case f (0) = 1.…”

“…We prove (2) in Proposition 1.2 by constructing a function ψ such that ψ (or its rotation) satisfies (2). First consider…”

Schwarz-Pick estimates for positive real part holomorphic functions on unit ball and polydisc

“…For notation, let D be the unit disk in C, B n and D n be the unit ball and polydisc in C n , respectively. There were many results on Schwarz-Pick estimates for holomorphic functions, see [1,3,6,8].…”

“…Motivated by [3], in this paper we obtain Carátheodory's inequality for positive real part holomorphic functions on the unit ball and polydisc; furthermore, estimates for higher order derivatives of all positive real part holomorphic functions on B n and D n are given. Before we give the main results, we recall some commonly used notations for multi-indices.…”

The high order Schwarz-Pick lemma on complex Hilbert balls

“…This has been generalized to the derivatives of arbitrary order [4,5,7]. The best result, proved in [4], is…”