## 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.

Show Summary Details# An improved Schwarz Lemma at the boundary

#### Open Access

## Abstract

## 1 Introduction

#### Lemma 1.1

## 2 Main result

#### Theorem 2.1

#### Proof

#### Remark 2.2

#### Remark 2.3

#### Remark 2.4

#### Remark 2.5

#### Remark 2.6

## 3 Consequences

#### Corollary 3.1

#### Proof

#### Remark 3.2

#### Remark 3.3

## References

## About the article

More options …# Open Mathematics

### formerly Central European Journal of Mathematics

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

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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

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

$$\underset{z\to x}{\mathrm{lim}\mathrm{inf}}\frac{1-|f(z)|}{1-\left|z\right|}=\beta .$$(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

$$\frac{|1-f(z){|}^{2}}{1-|f(z){|}^{2}}\le \beta \frac{|1-z{|}^{2}}{1-|z{|}^{2}}\text{\hspace{1em}}\forall z\in \Delta .$$

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 ∈ > *∂∆*:

$$\underset{r\nearrow 1}{\mathrm{lim}}\frac{f(r)-f(1)}{r-1}={f}^{\prime}(1)\text{\hspace{1em}}\text{\hspace{0.17em}with\hspace{0.17em}}\text{\hspace{1em}}|{f}^{\prime}(1)|=\beta .$$(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

$$\beta \ge 1+\frac{1-|{f}^{\prime}(0)|}{1+|{f}^{\prime}(0)|}.$$(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

$$\beta \ge \frac{2(1-\mathrm{Re}{f}^{\prime}(0))}{1-|{f}^{\prime}(0){|}^{2}}=1+\frac{|1-{f}^{\prime}(0){|}^{2}}{1-|{f}^{\prime}(0){|}^{2}}.$$(4)

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

$$\beta \ge \frac{2{(1-|f(0)|)}^{2}}{1-|f(0){|}^{2}+|{f}^{\prime}(0)|}.$$(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

$$|f(z)|\le \left|z\right|\text{\hspace{1em}}\forall z\in \Delta ,\text{\hspace{1em}}\text{\hspace{0.17em}and\hspace{0.17em}consequently\hspace{0.17em}}\text{\hspace{1em}}|{f}^{\prime}(0)|\le 1.$$

To remove the normalization *f*(0) = 0, one applies Schwarz’s Lemma to *ϕ*_{f(a)} ∘ *f* ∘ *ϕ _{a}* where

$${\varphi}_{a}(z)=\frac{a-z}{1-\overline{a}z}.$$

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

$$|\frac{f(w)-f(z)}{1-\overline{f(w)}f(z)}|\le |\frac{w-z}{1-\overline{w}z}|\text{\hspace{1em}}\forall z,w\in \Delta .$$

Consequently, the *hyperbolic derivative* satisfies

$$|{f}^{\ast}(z)|\le 1\text{\hspace{1em}}\forall z\in \Delta ,\text{\hspace{1em}}\text{where}{\text{f}}^{\ast}(\text{z})\text{=}\frac{{\text{1-|z|}}^{\text{2}}}{\text{1-|f}(\text{z}){\text{|}}^{\text{2}}}{\text{f}}^{\prime}(\text{z}).$$

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.

(Dieudonné’s Lemma). *Let f*: *∆* → *∆ be analytic, with f*(*z*) = *w and f*(*z*_{1}) = *w*_{1}. *Then*

$$|{f}^{\prime}(z)-c|\le r,$$

*where*

$$c=\frac{{\varphi}_{w}({w}_{1})}{{\varphi}_{z}({z}_{1})}\frac{1-|{\varphi}_{z}({z}_{1}){|}^{2}}{1-|{\varphi}_{w}({w}_{1}){|}^{2}}\frac{1-|w{|}^{2}}{1-|z{|}^{2}},r=\frac{|{\varphi}_{z}({z}_{1}){|}^{2}-|{\varphi}_{w}({w}_{1}){|}^{2}}{|{\varphi}_{z}({z}_{1}){|}^{2}(1-|{\varphi}_{w}({w}_{1}){|}^{2})}\frac{1-|w{|}^{2}}{1-|z{|}^{2}}.$$

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.

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

$$\beta \ge 2\frac{|1-w{|}^{2}}{1-|w{|}^{2}}\frac{1-|z{|}^{2}}{|1-z{|}^{2}}\frac{1-\mathrm{Re}({f}^{\ast}(z)\frac{1-\overline{w}}{1-w}\frac{1-z}{1-\overline{z}})}{1-|{f}^{\ast}(z){|}^{2}}.$$(6)

Using the easily verified identity

$$1-|{\varphi}_{a}(\lambda ){|}^{2}=\frac{(1-|a{|}^{2})(1-|\lambda {|}^{2})}{|1-\overline{a}\lambda {|}^{2}},$$(7)

we get, in Dieudonné’s Lemma,

$$c=\frac{{w}_{1}-w}{1-\overline{w}{w}_{1}}\frac{1-\overline{z}{z}_{1}}{{z}_{1}-z}\frac{1-|{z}_{1}{|}^{2}}{|1-\overline{z}{z}_{1}{|}^{2}}\frac{|1-\overline{w}{w}_{1}{|}^{2}}{1-|{w}_{1}{|}^{2}}=\frac{{w}_{1}-w}{{z}_{1}-z}\frac{1-w\overline{{w}_{1}}}{1-z\overline{{z}_{1}}}\frac{1-|{z}_{1}{|}^{2}}{1-|{w}_{1}{|}^{2}},$$

and

$$\begin{array}{cc}r& =\frac{\left(1-|{\varphi}_{w}({w}_{1}){|}^{2}\right)-\left(1-|{\varphi}_{z}({z}_{1}){|}^{2}\right)}{|{\varphi}_{z}({z}_{1}){|}^{2}(1-|{\varphi}_{w}({w}_{1}){|}^{2})}\frac{1-|w{|}^{2}}{1-|z{|}^{2}}\hfill \\ & =\frac{1}{|{\varphi}_{z}({z}_{1}){|}^{2}}(1-\frac{1-|z{|}^{2}}{1-|w{|}^{2}}\frac{1-|{z}_{1}{|}^{2}}{1-|{w}_{1}{|}^{2}}\frac{|1-\overline{w}{w}_{1}{|}^{2}}{|1-\overline{z}{z}_{1}{|}^{2}})\frac{1-|w{|}^{2}}{1-|z{|}^{2}}.\hfill \end{array}$$

then having *z*_{1} → 1 along a sequence for which *β* in (1) is attained, we get

$$c\to \text{\hspace{0.17em}}\tilde{c}={\left(\frac{1-w}{1-z}\right)}^{2}\frac{1}{\beta}\text{\hspace{1em}}\text{and}\text{\hspace{1em}}\text{r}\to \tilde{\text{\hspace{0.17em}r}}\text{=}\frac{{\text{1-|w|}}^{\text{2}}}{{\text{1-|z|}}^{\text{2}}}\text{-}\frac{\text{1}}{\beta}\frac{{\text{|1-w|}}^{\text{2}}}{{\text{|1-z|}}^{\text{2}}}.$$

That is,

$$|{f}^{\prime}(z)-\text{\hspace{0.17em}}\tilde{c}|\le \tilde{r}.$$(8)

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

$$|{f}^{\prime}(z){|}^{2}-2\mathrm{Re}\left(\overline{{f}^{\prime}(z)}{\left(\frac{1-w}{1-z}\right)}^{2}\frac{1}{\beta}\right)\le {\left({\scriptscriptstyle \frac{1-|w{|}^{2}}{1-|z{|}^{2}}}\right)}^{2}-\frac{2}{\beta}\frac{1-|w{|}^{2}}{1-|z{|}^{2}}\frac{|1-\overline{w}{|}^{2}}{|1-\overline{z}{|}^{2}}.$$

That is,

$${\left({\scriptscriptstyle \frac{1-|w{|}^{2}}{1-|z{|}^{2}}}\right)}^{2}(|{f}^{\ast}(z){|}^{2}-1)\le {\scriptscriptstyle \frac{2}{\beta}}{\scriptscriptstyle \frac{|1-\overline{w}{|}^{2}}{|1-\overline{z}{|}^{2}}}{\scriptscriptstyle \frac{1-|w{|}^{2}}{1-|z{|}^{2}}}[\mathrm{Re}({f}^{\ast}(z)\frac{1-\overline{w}}{1-w}\frac{1-z}{1-\overline{z}})-1].$$

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

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

$$\underset{r\nearrow 1}{\mathrm{lim}}{f}^{\prime}(r)={f}^{\prime}(1).$$

*From this, and using* $\left|\tau \right|=1\Rightarrow \frac{1-\mathrm{Re}(\sigma \tau )}{1-|\sigma {|}^{2}}\ge \frac{1}{1+\left|\sigma \right|}$, *follows the rather comforting fact that the right-hand side of (6) tends to β as z* → 1

*In Lemma 6.1 of [8] is the estimate*

$$\beta \ge \frac{2}{1+|{f}^{\ast}(z)|}\frac{1-|f(z)|}{1+|f(z)|}\frac{1-\left|z\right|}{1+\left|z\right|},$$(9)

*which contains (5), but is quite mild if* |*z*| or |*f*(*z*)| *is near* 1. *Anyway*, $\left|\tau \right|=1\Rightarrow \frac{1-\mathrm{Re}(\sigma \tau )}{1-\left|\sigma \right|}\ge 1$ *shows that* (6) *improves* (9).

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

$$\beta \ge 2\frac{|1-f(0){|}^{2}}{1-|f(0){|}^{2}}\frac{1-\mathrm{Re}({f}^{\ast}(0)\frac{1-\overline{f(0)}}{1-f(0)})}{1-{|{f}^{\ast}(0)|}^{2}}.$$(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*.

*Using again* $\left|\tau \right|=1\Rightarrow \frac{1-\mathrm{Re}(\sigma \tau )}{1-|\sigma {|}^{2}}\ge \frac{1}{1+\left|\sigma \right|}$ *in (10), we get*

$$\beta \ge \frac{2|1-f(0){|}^{2}}{1-|f(0){|}^{2}+|{f}^{\prime}(0)|},$$

*But using just* $\left|\tau \right|=1\Rightarrow \frac{1-\mathrm{Re}(\sigma \tau )}{1-|\sigma {|}^{2}}\ge \frac{1}{1+\mathrm{Re}(\sigma \tau )}$ *in (10), then* $\frac{1-|f(0){|}^{2}}{|1-f(0){|}^{2}}=\mathrm{Re}\frac{1+f(0)}{1-f(0)}$, *we get*

$$\beta \ge \frac{2}{\mathrm{Re}\frac{1-f{(0)}^{2}+{f}^{\prime}(0)}{{(1-f(0))}^{2}}},$$(11)

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

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.

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

$$\beta \ge 1+\frac{2|1-{f}^{\prime}(0){|}^{2}}{1-|{f}^{\prime}(0){|}^{2}+|{f}^{\prime \prime}(0)|/2}\frac{1+\mathrm{Re}\left(\frac{{f}^{\prime \prime}(0)}{2(1-|{f}^{\prime}(0){|}^{2})}\right)}{1-\frac{|{f}^{\prime \prime}(0)|}{2(1-|{f}^{\prime}(0){|}^{2})}}.$$(12)

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

$$g(\lambda )=\frac{f(\lambda )}{\lambda}\text{\hspace{0.17em}(with\hspace{0.17em}}g(0):={f}^{\prime}(0)\text{)},\text{\hspace{0.17em}and\hspace{0.17em}}\text{\hspace{1em}}h(\lambda )={\varphi}_{g(0)}(g(\lambda )).$$

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

$${h}^{\prime}(0)=\frac{-{f}^{\prime \prime}(0)}{2(1-|{f}^{\prime}(0){|}^{2})}.$$(13)

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

$$\underset{z\to 1}{\mathrm{lim}\mathrm{inf}}\frac{1-|h(z)|}{1-\left|z\right|}=(\beta -1)\frac{1-|{f}^{\prime}(0){|}^{2}}{|1-{f}^{\prime}(0){|}^{2}}=\widehat{\beta},\text{\hspace{0.17em}say}.$$(14)

Then in (6), i.e. (4), replacing *f* with *h* and *β* with $\widehat{\beta}$, we obtain

$$\beta \ge 1+\frac{|1-{f}^{\prime}(0){|}^{2}}{1-|{f}^{\prime}(0){|}^{2}}\frac{2(1-\mathrm{Re}{h}^{\prime}(0))}{1-|{h}^{\prime}(0){|}^{2}}.$$

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

*Corollary 3.1 improves*

$$\beta \ge 1+\frac{2(1-|{f}^{\prime}(0){|}^{2})}{1-|{f}^{\prime}(0){|}^{2}+|{f}^{\prime \prime}(0)|/2},$$(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).)*

*We add finally using that (4) in the form*

$$\beta \ge 1+\frac{|1-{f}^{\prime}(0){|}^{2}}{1-|{f}^{\prime}(0){|}^{2}},$$

*then replacing f with h and β with* $\widehat{\beta}$

$$\begin{array}{cc}\beta & \text{\hspace{0.17em}}\ge 1+\frac{|1-{f}^{\prime}(0){|}^{2}}{1-|{f}^{\prime}(0){|}^{2}}(1+\frac{|1+\frac{{f}^{\prime \prime}(0)}{2(1-|{f}^{\prime}(0){|}^{2})}{|}^{2}}{1-|\frac{{f}^{\prime \prime}(0)}{2(1-|{f}^{\prime}(0){|}^{2})}{|}^{2}})\hfill \\ & =1+\frac{|1-{f}^{\prime}(0){|}^{2}}{1-|{f}^{\prime}(0){|}^{2}}+\frac{|1+\frac{{f}^{\prime \prime}(0)}{2(1-|{f}^{\prime}(0){|}^{2})}{|}^{2}}{1-|\frac{{f}^{\prime \prime}(0)}{2(1-|{f}^{\prime}(0){|}^{2})}|}\frac{|1-{f}^{\prime}(0){|}^{2}}{1-|{f}^{\prime}(0){|}^{2}+|{f}^{\prime \prime}(0)|/2}.\hfill \end{array}$$

- [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

**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.

© 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

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