Article
Kyungpook Mathematical Journal 2020; 60(3): 507-518
Published online September 30, 2020
Copyright © Kyungpook Mathematical Journal.
Applications of the Schwarz Lemma and Jack's Lemma for the Holomorphic Functions
Bülent Nafi Örnek*, Batuhan Çatal
Department of Computer Engineering, Amasya University, Merkez-Amasya 05100, Turkey
e-mail : nafi.ornek@amasya.edu.tr
Department of Mathematics, Amasya University, Merkez-Amasya 05100, Turkey
e-mail : batuhancatal0591@gmail.com
Received: July 26, 2018; Revised: March 25, 2019; Accepted: April 23, 2019
We consider a boundary version of the Schwarz Lemma on a certain class of functions which is denoted by . For the function
Keywords: holomorphic function, the Schwarz lemma on the boundary, Jack's lemma, angular derivative.
1. Introduction
One of the main tool of complex functions theory is the Schwarz Lemma. This lemma is an important result which gives estimates about the values of holomorphic functions defined from the unit disc into itself. It plays an effective role in many fields of analysis, especially in the theory of geometric function hyperbolic geometry. The standard Schwarz Lemma, which is a direct application of the maximum modulus principle, is commonly stated as follows:
Let
We use the following lemma from [7] which is related to the function
Lemma 1.1.(Jack’s Lemma)
For historical background about the Schwarz Lemma and its applications on the boundary of the unit disc, we refer to [2, 6]. Also, a different application of Jack’s Lemma is shown in [7, 14, 18].
Let denote the class of functions
Let be a holomorphic function in the unit disc
where
From Jack’s Lemma, we have
Therefore, from (
Since
and
we take
and
This contradicts the assumed inequality (
The result is sharp and the extremal function is
Since the area of applicability of the Schwarz Lemma is quite wide, there exist many studies about it. Among these are the boundary version of the Schwarz Lemma, which is about estimating from below the modulus of the derivative of the function at some boundary point of the unit disc. The boundary version of the Schwarz Lemma is given as follows:
If
In addition to conditions of the boundary Schwarz Lemma, if
is obtained [19].
Inequality (
For our results, we need the following lemma called the Julia-Wolff Lemma (see [20]).
Lemma 1.2.(Julia-Wolff Lemma)
Corollary 1.3
In [5], all zeros of the holomorphic function in the unit disc different from
D. M. Burns and S. G. Krantz [3] and D. Chelst [4] studied the uniqueness part of the Schwarz Lemma. According to M. Mateljevic’s studies, some other types of results which are related to the subject can be found in [9, 10]. In addition, [11] was posed on ResearchGate where is discussed concerning results in more general aspects.
Mercer [12] prove a version of the Schwarz Lemma where the images of two points are known. Also, he considers some Schwarz and Carathéodory inequalities at the boundary, as consequences of a lemma due to Rogosinski [13].
In this work, we show an application of Jack’s Lemma for certain subclasses of holomorphic functions on the unit disc that provide (
2. Main Results
In this section, a boundary version of the Schwarz Lemma for holomorphic functions is investigated. The modulus of the angular derivative of the holomorphic function
Theorem 2.1
Let us consider the following function
where
Therefore, from (
and
Since
for
Therefore, we obtain
Now, we shall show that the inequality (
Then, we obtain
and
The inequality (
Theorem 2.2
Let
Since
and
we take
and
Now, we shall show that the inequality (
Then
On the other hand, from the Taylor expansion of
and
Therefore, we obtain
The inequality (
Theorem 2.3
Let
where
and
Furthermore, it can be seen that
Consider the function
This function is holomorphic in
Since
and
we take
and
Now, we shall show that the inequality (
Then
and
On the other hand, from the Taylor expansion of
and
Passing to limit in the last equality yields
Theorem 2.4
Let ,
Let
and
Take the following auxiliary function
It is obvious that Φ(
Since
and
we obtain
Also, since
and
we obtain
and
- TA. Azeroğlu, and BN. Örnek.
A refined Schwarz inequality on the boundary . Complex Var Elliptic Equ.,58 (2013), 571-577. - HP. Boas.
Julius and Julia: mastering the art of the Schwarz lemma . Amer Math Monthly.,117 (2010), 770-785. - DM. Burns, and SG. Krantz.
Rigidity of holomorphic mappings and a new Schwarz Lemma at the boundary . J Amer Math Soc.,7 (1994), 661-676. - D. Chelst.
A generalized Schwarz lemma at the boundary . Proc Amer Math Soc.,129 (2001), 3275-3278. - VN. Dubinin.
On the Schwarz inequality on the boundary for functions regular in the disc . J Math Sci.,122 (2004), 3623-3629. - GM. Golusin.
Geometric theory of functions of complex variable . Translations of Mathematical Monographs,26 , American Mathematical Society, Providence, R.I, 1969. - IS. Jack.
Functions starlike and convex of order α . J London Math Soc.,3 (1971), 469-474. - M. Jeong.
The Schwarz lemma and its application at a boundary point . J Korean Soc Math Educ Ser B: Pure Appl Math.,21 (2014), 219-227. - M. Mateljević.
Hyperbolic geometry and Schwarz lemma , Symposium MATHEMATICS AND APPLICATIONS, Faculty of Mathematics, University of Belgrade,VI 2015. - M. Mateljević.
Schwarz lemma, the Carathéodory and Kobayashi metrics and applications in cmplex analysis , XIX GEOMETRICAL SEMINAR, 2016, At Zlatibor,(), 1-12. - M. Mateljević.
. Rigidity of holomorphic mappings, Schwarz and Jack lemma., . - PR. Mercer.
Sharpened versions of the Schwarz lemma . J Math Anal Appl.,205 (1997), 508-511. - PR. Mercer.
Boundary Schwarz inequalities arising from Rogosinski’s lemma . J Class Anal.,12 (2018), 93-97. - R. Singh, and S. Singh.
Some sufficient conditions for univalence and starlikeness . Colloq Math.,47 (1982), 309-314. - BN. Örnek.
Sharpened forms of the Schwarz lemma on the boundary . Bull Korean Math Soc.,50 (2013), 2053-2059. - BN. Örnek.
Inequalities for the non-tangential derivative at the boundary for holomorphic function . Commun Korean Math Soc.,29 (2014), 439-449. - BN. Örnek.
Inequalities for the angular derivatives of certain classes of holomorphic functions in the unit disc . Bull Korean Math Soc.,53 (2016), 325-334. - BN. Örnek.
Estimates for holomorphic functions concerned with Jack’s lemma . Publ Inst Math.,104 (118)(2018), 231-240. - R. Osserman.
A sharp Schwarz inequality on the boundary . Proc Amer Math Soc.,128 (2000), 3513-3517. - Ch. Pommerenke. Boundary Behaviour of Conformal Maps,
, Springer-Verlag, Berlin, 1992.