Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


IMPACT FACTOR 2017: 0.831
5-year IMPACT FACTOR: 0.836

CiteScore 2018: 0.90

SCImago Journal Rank (SJR) 2018: 0.323
Source Normalized Impact per Paper (SNIP) 2018: 0.821

Mathematical Citation Quotient (MCQ) 2017: 0.32

ICV 2017: 161.82

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 16, Issue 1

Issues

Volume 13 (2015)

An improved Schwarz Lemma at the boundary

Peter R. Mercer
Published Online: 2018-10-19 | DOI: https://doi.org/10.1515/math-2018-0096

Abstract

We obtain an new boundary Schwarz inequality, for analytic functions mapping the unit disk to itself. The result contains and improves a number of known estimates.

Keywords: Schwarz Lemma; Julia’s Lemma

MSC 2010: 30C80

1 Introduction

Denote by ⊂ ℂ the open unit disk, and let f: be analytic. We assume that there is x > ∂∆ and β > ℝ such that

liminfzx1|f(z)|1|z|=β.(1)

By pre-composing with a rotation we may suppose that x = 1, and by post-composing with a rotation we may suppose that f(1) = 1. Then Julia’s Lemma (e.g. [1, 2]) gives

|1f(z)|21|f(z)|2β|1z|21|z|2zΔ.

This inequality has an appealing geometric interpretation, which we do not use here. But two immediate consequences which we do use, are that β > 0 and that the radial derivative of f exists at 1 ∈ > ∂∆:

limr1f(r)f(1)r1=f(1) with |f(1)|=β.(2)

(There are many other consequences of Julia’s Lemma, the most important being contained in the Julia-Carathéodory Theorems.)

Assuming the normalization f(0) = 0, we evidently have β ≥ 1. But even better, Osserman [3] showed that in this case

β1+1|f(0)|1+|f(0)|.(3)

(A proof of (3) can also be found in [4], which is motivated by the influential paper [5].) Now Osserman’s inequality was in fact anticipated by Ünkelbach [6], who had already obtained the better estimate

β2(1Ref(0))1|f(0)|2=1+|1f(0)|21|f(0)|2.(4)

However, [3] also contains a non-normalized version, which reduces to (3) if f(0) = 0, viz.

β2(1|f(0)|)21|f(0)|2+|f(0)|.(5)

Since the appearance of Osserman’s paper, a good number of authors have refined and generalized these estimates – as discussed in the next section. The aim here is to provide a different and very elementary approach, which contains and improves many of these modification. But first we recall some results which are of use in the sequel.

The well-known Schwarz’s Lemma, which is a consequence of the Maximum Principle, says that if f: is analytic with f(0) = 0, then

|f(z)||z|zΔ, and consequently |f(0)|1.

To remove the normalization f(0) = 0, one applies Schwarz’s Lemma to ϕf(a)fϕa where ϕa is the automorphism of which interchanges a and 0:

ϕa(z)=az1a¯z.

This gives the Schwarz-Pick Lemma which says that for f: analytic,

|f(w)f(z)1f(w)¯f(z)||wz1w¯z|z,wΔ.

Consequently, the hyperbolic derivative satisfies

|f(z)|1zΔ,wheref(z)=1-|z|21-|f(z)|2f(z).

It is the Schwarz-Pick Lemma that does most of the work in proving Julia’s Lemma. But another consequence of the Schwarz-Pick Lemma is the following (e.g. [7–9]), which we shall also rely upon.

Lemma 1.1

(Dieudonné’s Lemma). Let f: ∆ be analytic, with f(z) = w and f(z1) = w1. Then

|f(z)c|r,

where

c=ϕw(w1)ϕz(z1)1|ϕz(z1)|21|ϕw(w1)|21|w|21|z|2,r=|ϕz(z1)|2|ϕw(w1)|2|ϕz(z1)|2(1|ϕw(w1)|2)1|w|21|z|2.

2 Main result

We remove the dependence on f(0), while improving many estimates which do contain f(0). We shall rely on Dieudonné’s Lemma, the Schwarz-Pick Lemma, and Julia’s Lemma.

Theorem 2.1

Let f: ∆ be analytic with f(z) = w and f(1) = 1 as in (1). Then

β2|1w|21|w|21|z|2|1z|21Re(f(z)1w¯1w1z1z¯)1|f(z)|2.(6)

Proof

Using the easily verified identity

1|ϕa(λ)|2=(1|a|2)(1|λ|2)|1a¯λ|2,(7)

we get, in Dieudonné’s Lemma,

c=w1w1w¯w11z¯z1z1z1|z1|2|1z¯z1|2|1w¯w1|21|w1|2=w1wz1z1ww1¯1zz1¯1|z1|21|w1|2,

and

r=(1|ϕw(w1)|2)(1|ϕz(z1)|2)|ϕz(z1)|2(1|ϕw(w1)|2)1|w|21|z|2=1|ϕz(z1)|2(11|z|21|w|21|z1|21|w1|2|1w¯w1|2|1z¯z1|2)1|w|21|z|2.

then having z1 → 1 along a sequence for which β in (1) is attained, we get

cc˜=(1w1z)21βandr r˜=1-|w|21-|z|2-1β|1-w|2|1-z|2.

That is,

|f(z)c˜|r˜.(8)

Now, upon squaring both sides of this inequality, there is some cancellation:

|f(z)|22Re(f(z)¯(1w1z)21β)(1|w|21|z|2)22β1|w|21|z|2|1w¯|2|1z¯|2.

That is,

(1|w|21|z|2)2(|f(z)|21)2β|1w¯|2|1z¯|21|w|21|z|2[Re(f(z)1w¯1w1z1z¯)1].

By the Schwarz-Pick Lemma each side of this last inequality is nonpositive, so isolating β we get (6).

Remark 2.2

Having z → 1 radially in line (8), and using (2), we obtain

limr1f(r)=f(1).

From this, and using |τ|=11Re(στ)1|σ|211+|σ|, follows the rather comforting fact that the right-hand side of (6) tends to β as z → 1 radially.

Remark 2.3

In Lemma 6.1 of [8] is the estimate

β21+|f(z)|1|f(z)|1+|f(z)|1|z|1+|z|,(9)

which contains (5), but is quite mild if |z| or |f(z)| is near 1. Anyway, |τ|=11Re(στ)1|σ|1 shows that (6) improves (9).

Remark 2.4

Now take z = 0, so that (6) reads

β2|1f(0)|21|f(0)|21Re(f(0)1f(0)¯1f(0))1|f(0)|2.(10)

This may be regarded as an non-normalized version of (4). Indeed, taking also f(0) = 0 recovers (4). This is the same estimate which results from having z = 0 in Theorem 5 of [10]. However, that result (which is arrived at by very nonelementary means) contains f(0) even for z ≠ 0, a deficiency from which Theorem 2.1 does not suffer.

Remark 2.5

Using again |τ|=11Re(στ)1|σ|211+|σ| in (10), we get

β2|1f(0)|21|f(0)|2+|f(0)|,

which improves (5), analogously to how (4) improves (3).

Remark 2.6

But using just |τ|=11Re(στ)1|σ|211+Re(στ) in (10), then 1|f(0)|2|1f(0)|2=Re1+f(0)1f(0), we get

β2Re1f(0)2+f(0)(1f(0))2,(11)

which improves (5) more effectively. Estimate (11) was obtained differently in each of [11] and [12].

3 Consequences

Cases for which z = w = 0 (i.e. f(0) = 0) are obviously contained in the remarks above, but when this holds we can do a little better, as follows.

Corollary 3.1

Let f: ∆ be analytic with f(0) = 0 and f(1) = 1 as in (1). Then

β1+2|1f(0)|21|f(0)|2+|f(0)|/21+Re(f(0)2(1|f(0)|2))1|f(0)|2(1|f(0)|2).(12)

Proof

We introduce f″(0), in standard fashion: Set

g(λ)=f(λ)λ (with g(0):=f(0)), and h(λ)=ϕg(0)(g(λ)).

Then h is analytic on with h(0) = 0, and by Schwarz’s Lemma h: . Here we have

h(0)=f(0)2(1|f(0)|2).(13)

A calculation using the identity (7) and the assumption (1) gives

liminfz11|h(z)|1|z|=(β1)1|f(0)|2|1f(0)|2=β^, say.(14)

Then in (6), i.e. (4), replacing f with h and β with β^, we obtain

β1+|1f(0)|21|f(0)|22(1Reh(0))1|h(0)|2.

Inserting (13) and a little tidying yields (12), as desired.

Remark 3.2

Corollary 3.1 improves

β1+2(1|f(0)|2)1|f(0)|2+|f(0)|/2,(15)

which was obtained by Dubinin [13] using a proof which relies directly on (3). (Incidentally, Schwarz’s Lemma applied to h gives |f″(0)|/2 ≤ 1 − |f′(0)|2, from which it is readily seen that (15) improves (3).)

Remark 3.3

We add finally using that (4) in the form

β1+|1f(0)|21|f(0)|2,

then replacing f with h and β with β^ here, and using (13) and (14), we get another way of expressing (12):

β1+|1f(0)|21|f(0)|2(1+|1+f(0)2(1|f(0)|2)|21|f(0)2(1|f(0)|2)|2)=1+|1f(0)|21|f(0)|2+|1+f(0)2(1|f(0)|2)|21|f(0)2(1|f(0)|2)||1f(0)|21|f(0)|2+|f(0)|/2.

References

  • [1]

    Carathéodory C., Theory of Functions, Vol II, 1960, New York: Chelsea. Google Scholar

  • [2]

    Cowen C.C., Maccluer B.D., Composition Operators on Spaces of Analytic Functions, 1995, Boca Raton: C.R.C. Press. Google Scholar

  • [3]

    Osserman R., A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc., 2000, 128, 3513-3517. CrossrefGoogle Scholar

  • [4]

    Krantz S.G., The Schwarz lemma at the boundary, Complex Var. & Ellip. Equ., 2011, 56, 455-468. CrossrefGoogle Scholar

  • [5]

    Burns D.M., Krantz S.G., Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary, J. Amer. Math. Soc., 1994, 7, 661-676. CrossrefGoogle Scholar

  • [6]

    Ünkelbach H., Uber die Randverzerrung bei konformer Abbildung, Math. Zeit., 1938, 43, 739-742. CrossrefGoogle Scholar

  • [7]

    Duren P.L., Univalent Functions, 1983, New York & Berlin: Springer-Verlag.Google Scholar

  • [8]

    Kaptanoğlu H.T., Some refined Schwarz-Pick Lemmas, Mich. Math. J., 2002, 50, 649-664. CrossrefGoogle Scholar

  • [9]

    Mercer P.R., Sharpened versions of the Schwarz lemma, J. Math. Analysis & Appl., 1997, 205, 508-511.CrossrefGoogle Scholar

  • [10]

    Komatu Y., On angular derivative, Kodai Math. Sem. Rep., 1961, 13, 167-179. CrossrefGoogle Scholar

  • [11]

    Frovlova A., Levenshtein M., Shoikhet D., Vasil’ev A., Boundary distortion estimates for holomorphic maps, Complex Anal. Oper. Theory, 2014, 8, 1129-1149. CrossrefWeb of ScienceGoogle Scholar

  • [12]

    Ren G., Wang X., Extremal functions of boundary Schwarz lemma. arXiv.org. https://arxiv.org/pdf/1502.02369

  • [13]

    Dubinin V.I., Schwarz inequality on the boundary for functions regular in the disk, J. Math. Sci, 2004, 122, 3623-3629. CrossrefGoogle Scholar

About the article

Received: 2018-04-26

Accepted: 2018-09-05

Published Online: 2018-10-19


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1140–1144, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0096.

Export Citation

© 2018 Mercer, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Comments (0)

Please log in or register to comment.
Log in