Schwarz lemma involving the boundary fixed point

R E S E A R C H

Open Access

Schwarz lemma involving the boundary

fixed point

Qinghua Xu

_{, Yongfa Tang}

_{, Ting Yang}

_{and Hari Mohan Srivastava}

*_{Correspondence:} xuqh@mail.ustc.edu.cn 1_{College of Mathematics and} Information Science, JiangXi Normal University, NanChang, 330022, People’s Republic of China Full list of author information is available at the end of the article

Abstract

Letfbe an holomorphic function which maps the unit disk into itself. In this paper, consider the zero of orderk(i.e.,f(z) –f(0) (orf(z)) has a zero of orderkatz= 0), we obtain the sharp estimates of the classical boundary Schwarz lemma involving the boundary fixed point. The results presented here would generalize the corresponding result obtained by Frolovaet al.(Complex Anal. Oper. Theory 8:1129-1149, 2004).

MSC: 30C45; 32A10

Keywords: boundary Schwarz lemma; fixed point; zero of order

1 Introduction and preliminaries

It is well known that the Schwarz lemma serves as a very powerful tool to study several re-search fields in complex analysis. For example, almost all results in the geometric function theory have the Schwarz lemma lurking in the background [–].

On the other hand, Schwarz lemma at the boundary is also an active topic in complex analysis, various interesting results have been obtained [–]. Before summarizing these results, it is necessary to give some elementary contents on the boundary fixed points []. LetDdenote the unit disk inC,H(D,D) denote the class of holomorphic self-mappings ofD,Ndenote the set of all positive integers. The boundary pointξ∈∂Dis called a fixed point off ∈H(D,D) if

f(ξ) = lim

r→–f(rξ) =ξ.

The classification of the boundary fixed points off ∈H(D,D) can be performed via the value of the angular derivative

f(ξ) =∠lim

z→ξ f(z) –ξ

z–ξ ,

which belongs to (,∞] due to the celebrated Julia-Carathédory theorem []. This the-orem also asserts that the finite angular derivative at the boundary fixed pointξ exists if and only if the holomorphic functionf(z) has the finite angular limit∠limz→ξf(z). For a

boundary fixed pointξoff, if

f(ξ)∈(,∞),

thenξis called a regular boundary fixed point. The regular fixed points can be attractive iff(ξ)∈(, ), neutral iff(ξ) = , or repulsive iff(ξ)∈(,∞).

By the Julia-Carathédory theorem [] (see also []) and the Wolff lemma [], iff ∈ H(D,D) with no interior fixed point, then there exists a unique regular boundary fixed pointξsuch thatf(ξ)∈(, ]; and iff ∈H(D,D) with an interior fixed point, thenf(ξ) >  for any boundary fixed pointξ∈∂D.

In particular, Unkelbach [] (see also []) obtain the following boundary Schwarz lemma.

Theorem A If f ∈H(D,D)has a regular boundary fixed point,and f() = ,then

f()≥ 

 +|f()|. ()

Moreover,equality in()holds if and only if f is of the form

f(z) = –z a–z

 –az, ∀z∈D,

for some constant a∈(–, ].

Theorem A was improved  years later by Osserman [] by removing the assumption

f() = .

Theorem B([]) If f ∈H(D,D)withξ= as its regular boundary fixed point.Then

f()≥ ( –|f()|)



 –|f()|₊|_f()|. ()

In [], Frolovaet al.proved the following theorem, which is an improvement of Theo-rem B.

Theorem C([]) If f ∈H(D,D)withξ= as its regular boundary fixed point.Then

f()≥ 

e(–f_(–()_f_())+f())

. ()

Recently, Ren and Wang [] offered an alternative and elementary proof of Theorem C and studied the extremal functions of the inequality (). Their method of proof is quite different from that which Frolovaet al.have used in [].

In this paper, stimulated by the above-cited work (especially []), considering the zero of order, we obtain a version of boundary Schwarz lemma. This result is a generalization of the boundary Schwarz-Pick lemma obtained by Frolovaet al.[].

In order to prove the desired results, we first recall the classical Julia lemma [] and the Julia-Carathéodory theorem [].

Lemma  ([]) Let f ∈H(D,D) and letξ ∈∂D. Suppose that there exists a sequence {zn}n∈N⊂Dconverging toξ as n tends to∞,such that the limits

α= lim

n→∞

 –|f(zn)|

and

η= lim

n→∞f(zn)

exist(finitely).Thenα> and the inequality

|f(z) –η|

 –|f(z)| ≤α

|z–ξ|

 –|z| ()

holds throughout the open unit disk Dand is strict except for Möbius transformations ofD.

Lemma ([]) Let f ∈H(D,D)and letξ∈∂D.Then the following conditions are equiv-alent:

(i) The lower limit

α=lim inf

z→ξ

 –|f(z)|

 –|z| ()

is finite,where the limit is taken aszapproachesξunrestrictedly inD;

(ii) f has a non-tangential limit,sayf(ξ),at the pointξ,and the difference quotient

f(z) –f(ξ)

z–ξ

has a non-tangential limit,sayf(ξ),at the pointξ;

(iii) the derivativefhas a non-tangential limit,sayf(ξ),at the pointξ.Moreover,

under the above conditions we have: (a) αin(i);

(b) the derivativesf(ξ)in(ii)and(iii)are the same; (c) f(ξ) =αξf(ξ);

(d) the quotient–_–|f_|z|(z)|has the non-tangential limitα,at the pointξ.

Lemma ([], p.) Letϕ∈H(D,D),andϕ(z) =∞_n=bnzν.Then

|bn| ≤ –|b|, n≥.

2 Main results and their proofs

We now state and prove each of our main results given by Theorems  and  below.

Theorem  Let f ∈H(D,D)withξ = as its regular boundary fixed point and suppose f() =f() =· · ·=f(k–)_{() = ,a}

k=f

(k)_()

k! = ,k∈N,we can obtain:

(I) if <|ak|< ,then

f()≥k+| –ak|



 –|ak|

  +e–ak

–ak ak+

–|ak|

whereak+=f

(k+)_()

(k+)! .Equality holds in the inequality if and only iff is of the form

f(z) =zk ak–z

z–a

–az ak–

–ak  –z_–z–_aza ak–

–akak

, ∀z∈D, ()

for some constanta∈[–, ). (II) If|ak|= ,thenf(z) =zk.

Proof In view of Lemma , we consider the following two cases.

Case IIf  <|ak|< , let

g(z) = ⎧ ⎪ ⎨ ⎪ ⎩

–ak ak–

ak–f(z) zk

–akf_zk(z)

,  <|z|< ,

, z= .

It is elementary to see that g∈H(D,D), andξ =  is its regular boundary fixed point. A straightforward computation shows that

f() =k+| –ak|



 –|ak|

g() ()

and

g() = –ak  –ak ·

ak+

 –|ak|

, ()

which is no larger than  in modulus. Applying Lemmas  and  to the holomorphic func-tionh:D→Ddefined by

h(z) = g(z)

z , ∀z∈D,

we obtain

g() =  +h()≥ +| –g

_()|

 –|g()|=

( –eg())

 –|g()| . ()

In particular,

g()≥ 

 +eg(). ()

By combining (), (), and (), we get the estimate in ().

Furthermore, this bound in () is sharp. Indeed, if equality holds in () forz∈D, then we must have equalities in the corresponding inequalities in () and (). Thus, we can obtain

g(z) =zz–a

 –az

 –a

 –a ()

Consequently,f must be of the form

f(z) =zk ak–z

z–a

–az ak–

–ak  –z_–z–_aza ak–

–akak

, ∀z∈D, ()

for some constanta∈[–, ).

Case IIIf|ak|= , set

g(z) = ⎧ ⎨ ⎩

f(z)

zk ,  <|z|< ,

ak, z= .

It is clear thatg∈H(D,D),|g()|=|ak|= . Thus by the principle of the maximum

modu-lus,gis a constant function, andg(z) =ak=g() = , and hencef(z)≡zk. This completes

the proof.

Taking into account the relation|–ak

–ak · ak+

–|ak|| ≤ and using () in Theorem , we can readily deduce the following corollary (the proof is omitted here).

Corollary  Let f ∈H(D,D)withξ= as its regular boundary fixed point and suppose f() =f() =· · ·=f(k–)_{() = ,a}

k=f

(k)_()

k! = ,k∈N;we have the following.

If <|ak|< ,then

f()≥k+| –ak|



 –|ak|

. ()

In particular,

f()≥k–  +   +eak

. ()

Remark  Whenn= , it follows from () that

f()≥   +ea

= 

 +ef().

Note that

  +ef()≥

  +|f()|.

Therefore, Theorem  (or Corollary ) generalizes and improves Theorem A.

Theorem  Let f ∈H(D,D)withξ= as its regular boundary fixed point and suppose f() =· · ·=f(k–)() = ,ak=f

(k)_()

k! = ,k∈N,we can obtain:

(I) If <|ak|<  –|f()|,then

f()≥(k– )| –f()|



 –|f()| +



e(–f()+ak

(–f()) )

Equality holds in the inequality if and only iff is of the form

f(z) =

f() –zk a–z

–az

–f() –f()

 –zk a–z

–az

–f() –f()f()

.

(II) If|ak|=  –|f()|,then

f(z) =

–f() –f()z

k_+f()

 +f()–f()

–f()z

k. ()

Proof Set

g(z) = f(z) –f()  –f()f(z)

 –f()  –f().

It is not difficult to verify thatg∈H(D,D), andξ=  is its regular boundary fixed point. Elementary computations yield

f() = | –f()|



 –|f()|g

_{() ()}

and

g(k)_()

k! =

ak

 –|f()|

 –f()

 –f(). ()

On the other hand, let

h(z) = ⎧ ⎨ ⎩

g(z)

zk ,  <|z|< , g(k)()

k! , z= ,

()

which is inH(D,D). By Lemma , we obtain the following results: (I) If  <|ak|<  –|f()|, then it follows from () that|g

(k)_()

k! |< . By using Lemmas 

and , we have

g() =k+h()≥k+| –

g(k)()

k! |

 –|g(k_k)_!()| =k–  +

( –eg(k_k)_!())  –|g(k_k)()_! | .

In particular,

g()≥k–  +   +eg(k_k)_!()

.

From the above relation and (), we deduce that

f()≥| –f()|



 –|f()|

k–  + 

 +eg(k_k)()_!

=| –f()|



 –|f()|

k–  + 

 +e( ak

–|f()|

–f() –f())

= (k– )| –f()|



 –|f()| +



e(–f()+ak

(–f()) )

.

Applying a similar argument to Theorem , we deduce that equality holds in inequality () if and only iff is of the form

f(z) =

f() –zk a_––_azz –f()

–f()

 –zk a–z

–az

–f() –f()f()

.

(II) If|ak|=  –|f()|, then we find from () and () that|h()|=|g

(k)_()

k! |= . By the

principle of the maximum modulus,his a constant function, andh(z) =g() = , and hence

g(z)≡zk, which yields the assertion (). This completes the proof.

Remark  By settingk=  in () of Theorem , we get the following estimate:

f()≥ 

e(–f()+a

(–f()) )

= 

e(–_(–f()_f_())+f())

,

which is Theorem C obtained by Frolovaet al.[]. Thus, Theorem  is a generalization of Theorem C.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally to this work.

Author details

1_{College of Mathematics and Information Science, JiangXi Normal University, NanChang, 330022, People’s Republic of} China.2_{Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3P4, Canada.} 3_{China Medical University, Taichung, 40402, Taiwan, People’s Republic of China.}

Acknowledgements

This work was supported by NNSF of China (Grant Nos. 11561030, 11261022), the Jiangxi Provincial Natural Science Foundation of China (Grant No. 20152ACB20002), and Natural Science Foundation of Department of Education of Jiangxi Province, China (Grant No. GJJ150301).

Received: 16 March 2016 Accepted: 4 August 2016

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