A Refinement of Schwarz–Pick Lemma for Higher Derivatives

Ern Kwon,Mi Kim,Gun Kwon,Jinkee Lee

doi:10.3390/math7010077

Ern Kwon, Mi Kim + Show 2 more

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https://doi.org/10.3390/math7010077

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Abstract

In this paper, a Schwarz–Pick estimate of a holomorphic self map f of the unit disc D having the expansion f ( w ) = c 0 + c n ( w − z ) n + … in a neighborhood of some z in D is given. This result is a refinement of the Schwarz–Pick lemma, which improves a previous result of Shinji Yamashita.

Highlights

For the open unit disc D of the complex plane and the boundary ∂D of D, the followingSchwarz–Pick lemma(see [1], Lemma 1.2) is well-known.Theorem 1
We presented Schwarz-Pick Lemma for higher derivatives in connection with p-mean M p (r, f )(see Theorem 4). It refined a previous result of Shinji Yamashita and clarified the condition of equality
Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2017R1E1A1A03070738)

Summary

Introduction

For the open unit disc D of the complex plane and the boundary ∂D of D, the following. 1 + āz φ a satisfies φ a ( φ− a (z)) = z for all z ∈ D It is well-known that φ a (∂D ) = ∂D and that the set of automorphisms, i.e., bijective biholomorphic mappings, of D consists of the mappings of the form αφ a (z), where a ∈ D and |α| = 1. Let f be a function holomorphic and bounded, | f | < 1, in D and let 0 ≤ p ≤ ∞. It is expected that there might be a refinement of Theorem 2 which reduces to Theorem 3 when n = 1 This is our objective of this note

Result

Proof of Theorem 4

Applications

Conclusions