Some Sharp Schwarz Inequalities of the Unit Disk in

WU Keke, CHEN Wei, TANG Xiaomin

(School of Science, Huzhou University, Huzhou 313000, China)

SomeSharpSchwarzInequalitiesoftheUnitDiskin

WU Keke, CHEN Wei, TANG Xiaomin

(School of Science, Huzhou University, Huzhou 313000, China)

In this paper, we establish a new type of the classical Schwarz lemma for holomorphic functions on the unit disk in, and we also obtain some Schwarz inequalities of the derivative of the holomorphic function at a boundary point of the unit disk. Those results extend the classical inner Schwarz lemma and boundary Schwarz lemma respectively.

holomorphic function; Schwarz lemma; unit disk; boundary point

0 Introduction

The Schwarz lemma is one of the most important results in the classical complex analysis, which has become a crucial theme in many branches of mathematical research for over a hundred years. LetDbe the open unit disk in the complex plane. And let ?Dbe the boundary ofD. The classical Schwarz lemma is stated as follows.

Theorem1[1]Letf:D→Dbe holomorphic andf(0)=0. Then |f(z)|≤|z| for anyz∈Dand |f′(0)|≤1. Moreover, if |f(z0)|=|z0| for somez0∈D{0} or if |f′(0)|=1, then there is a real numberθsuch thatf(z)=eiθz.

A great deal of work has been devoted to generalizations of Schwarz lemma to more general settings. We refer to [2-7] for a more complete insight on the Schwarz lemma.

From the point of view of applications, it has been a very natural task to obtain various versions of the Schwarz lemma at the boundary. There is the following classical boundary Schwarz lemma.

Theorem2[1]Letf:D→Dbe a holomorphic function. Iffis holomorphic atz=1 withf(0)=0 andf(1)=1, thenf′(1)≥1. Moreover, the inequality is sharp.

In [8], ?rnek got some sharp forms of the Schwarz lemma on the boundary of the unit disk. Osserman gave the Schwarz lemma at the boundary of the unit disk, and presented some sharp Schwarz inequalities at a boundary point of the unit disk in [9]. One of the typical results, in this paper, is stated as follows.

Theorem3[9]Letf:D→Dbe a holomorphic function. Iff(z) is holomorphic atz0∈?Dwithf(0)=0 and |f(z0)|=1, then

Moreover,

|f′(z0)|≥1.

(1)

And the equality holds in (1) if and only iff(z)=zeiθf(wàn)or someθ∈. Furthermore, iff(0)=f′(0)=…=f(n-1)(0)=0, then

|f′(z0)|≥n.

(2)

The equality holds in (2) if and only iff(z)=zneiθf(wàn)or someθ∈.

Establishing various versions of the Schwarz lemma at the boundary has attracted attentions of many mathematicians. Here we refer the reader to [10-15], as well as, many references therein for discussions related to such studies. Our main purpose here is to establish a new type of the classical Schwarz lemma for holomorphic function on the unit disk, and give the optimal estimates of the derivative of the holomorphic function at a boundary point of the unit disk.

1 Main results

We first introduce some notations and definitions, and present the Schwarz lemma for the holomorphic functions on the unit disk.

Leta∈Dand consider the M?bius mappingφaofDthat interchangesaand 0,

Theorem4 Letfbe a holomorphic function onDwith |f(z)-b|<1 andf(0)=a, where -1+b

(3)

Moreover,

|f′(0)|≤1-(a-b)2.

(4)

The equality in (3) for some nonzeroz∈Dor in (4) holds if and only if

for someθ∈.

ProofTakeg(z)=f(z)-band

for anyz∈D. Thengandφare both holomorphic self-mappings ofDwithφ(0)=0. It follows from Theorem 1 that |φ(z)|≤|z| for eachz∈Dand |φ′(0)|≤1. This implies

Hence, we obtain

|f(z)|-a≤|f(z)-a|≤|z||1-(a-b)(f(z)-b)|≤|z|(|1+(a-b)b|+|a-b||f(z)|).

This gives

Notice that

This, together with |φ′(0)|≤1, yields

Thus, we have

|f′(0)| ≤1-(a-b)2.

By Theorem 1, the equality in (3) for some nonzeroz∈Dor in (4) holds if and only if

whereθ∈. It follows that

whereθ∈. The proof is complete.

Now, we give the Schwarz inequality of holomorphic function at a boundary point of the unit disk.

Theorem5 Letfbe a holomorphic function onDwith |f(z)-b|<1 andf(0)=a, where -1+b

(5)

The equality holds in (5) if and only if

whereθ∈satisfieseiθ=.

Thus, by the Theorem 3 we get

|φ′(z0)|≥1.

(6)

Since

(7)

combine (6) and (7) to obtain

It follows that

Notice that

Then

[1+(a-b)b]z0eiθ+a= [1+(a-b)z0eiθ](1 +b),

This yields

The proof is complete.

Finally, we consider the Schwarz inequality at a boundary point of the unit disk for the holomorphic function with some special Taylor expansion.

Theorem6 Letfbe a holomorphic function onD. Suppose thatf(z)=a+cnzn+cn+1zn+1+… and |f(z)-b|<1, wheren≥1,cn≠0, -1+b

(8)

The equality holds in (8) if and only if

whereθ∈satisfieseiθ=.

bnzn+bn+1zn+1+…,

wherebn,bn+1，…∈andbn=≠0. Hence,

φ(0)=φ′(0)=φ″(0)=…=φn-1(0)=0.

It follows from Theorem 3 that

This implies

If the equality holds in (8), then |φ′(z0) |=n. Hence, by Theorem 3 we have

which yields

whereθ∈satisfieseiθ, Then

Thus, we have

The proof is complete.

RemarkFrom the proof of Theorem 5 and 6, it is clear that we only to need assume that the function f is1up to the boundary ofDnearz0.

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單位圓盤(pán)上若干精確的Schwarz不等式

吳科科, 陳 偉, 唐笑敏

(湖州師范學(xué)院 理學(xué)院， 浙江 湖州 313000)

建立了復(fù)平面中單位圓盤(pán)上全純函數(shù)的一個(gè)新型Schwarz引理，并獲得了單位圓盤(pán)上全純函數(shù)的導(dǎo)函數(shù)在邊界點(diǎn)處的若干Schwarz不等式.這些結(jié)果分別推廣了經(jīng)典的內(nèi)部型Schwarz引理和邊界型Schwarz引理.

全純函數(shù); Schwarz引理; 單位圓盤(pán); 邊界點(diǎn)

O174.56

date：2017-09-14

s:This work is supported by the NNSF of China (11571105) and the Xinmiao Talent Project of Zhejiang Province (2016R427003).

Biography:TANG Xiaomin, Ph.D, Professor, Research Interests: complex analysis. E-mail:txm@zjhu.edu.cn

O174.56DocumentcodeAArticleID1009-1734(2017)10-0006-06

MSC2010:30C80； 32H02

MSC2010:30C80； 32H02

[責(zé)任編輯吳志慧]