### 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 _{2}^{2} + _{3}^{3} + … which is defined in the unit disc

**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 ^{iσ} for real

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 _{2}^{2} + _{3}^{3} + … which are holomorphic in

Let be a holomorphic function in the unit disc

where _{0} ∈

From Jack’s Lemma, we have

Therefore, from (

Since

and

we take

and

This contradicts the assumed inequality (_{0} ∈ _{0})| = 1 for all

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 (^{iθ},

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

**Proof**

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 (_{2} which is second coefficient in the expansion of the function _{2}^{2}+_{3}^{3}+….

### Theorem 2.2

**Proof**

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 (_{3} which is third coefficient in the expansion of the function _{2}^{2}+_{3}^{3}+….

### Theorem 2.3

**Proof**

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 _{2} = 0. Similarly, using straightforward calculations, we take

### Theorem 2.4

Let , _{2} > 0

**Proof**

Let _{2} > 0 and let us consider the function _{2}|, we denote by ln

and

Take the following auxiliary function

It is obvious that Φ(

Since

and

we obtain

Also, since

and

we obtain

and

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