# Univalent functions in the Banach algebra of continuous functions

- Yong Chan Kim
^{1}and - Jae Ho Choi
^{2}Email author

**2013**:145

https://doi.org/10.1186/1029-242X-2013-145

© Kim and Choi; licensee Springer 2013

**Received: **24 December 2012

**Accepted: **7 March 2013

**Published: **2 April 2013

## Abstract

In this paper, we investigate several interesting properties of a composition operator defined on the open unit ball ${B}_{0}$ of the Banach algebra $C(T)$. We also consider the Noshiro-Warschawski theorem in the Banach algebra of continuous functions.

**MSC:**30C45, 46J10.

## Keywords

## 1 Introduction and definitions

Throughout this paper, $C(T)$ denotes the Banach algebra, with sup norm, of continuous complex-valued functions defined on a compact metric space *T*. Let $B(f:r)$ be an open ball in $C(T)$ centered at $f\in C(T)$ with radius *r*. In particular, for the sake of brevity, we use the simplified notation ${B}_{0}$ instead of $B(0:1)$.

*univalent*in the unit disk $\mathcal{U}$. A function $\phi (z)$ belonging to the class $\mathcal{S}$ is said to be

*convex*in $\mathcal{U}$ if and only if

We denote by $\mathcal{K}$ the class of all functions in $\mathcal{S}$ which are convex in $\mathcal{U}$.

*φ*. In this case, we cannot ensure the convergence of the series

*G*be an open nonempty subset of $C(T)$. A function $F:G\to C(T)$ is said to be

*L*-

*differentiable*at a point $f\in G$ if there exists $\lambda \in C(T)$ and a map

*η*defined in a ball $B(0:r)$ with values in $C(T)$ such that

*λ*the

*L*-

*derivative*of

*F*at

*f*and denote it by ${F}^{\mathrm{\prime}}(f)$. From [1], we see that

where ${\phi}^{\mathrm{\prime}}$ is a derivative of *φ*.

In the present paper, we investigate several geometric properties of the class ${S}_{C}$ associated with the theory of univalent functions.

## 2 Geometric properties of the composition operator ${F}_{\phi}$

We begin by proving the following theorem.

**Theorem 1** ${F}_{\phi}\in {S}_{C}$ *if and only if* $\phi \in \mathcal{S}$.

*Proof*(⇐) Suppose that ${F}_{\phi}(f)={F}_{\phi}(g)$ for the functions

*f*and

*g*in ${B}_{0}$. Then it means that

for all $t\in T$. Since *φ* is univalent, $f(t)=g(t)$ for all $t\in T$.

*f*and

*g*such that $f={z}_{1}$ and $g={z}_{2}$, then it is obvious that

Since ${F}_{\phi}$ is injective, we have $f=g$. Hence we get ${z}_{1}={z}_{2}$. This completes the proof of Theorem 1. □

By using Brange’s theorem [2], we obtain the following.

**Corollary 1**

*If*

*then*

Now we prove the Noshiro-Warschawski theorem ([3], Theorem 2.16) in the Banach algebra $C(T)$.

**Theorem 2**

*If the*

*L*-

*derivative*${F}_{\phi}^{\mathrm{\prime}}(f)$

*has a positive real part for all*$f\in {B}_{0}$,

*then*

*Proof*If ${f}_{1}\in {B}_{0}$, ${f}_{2}\in {B}_{0}$ and ${f}_{1}\ne {f}_{2}$, then there exists $t\in T$ such that

at $t\in T$, which shows that ${F}_{\phi}$ is injective. □

**Remark**Since

*T*is compact, $\{f(t):t\in T\}$ is a closed proper subset of $\mathcal{U}$. Hence the condition (2.2) does not imply

Next we obtain the following.

**Theorem 3**

*Let*

*Then*

*is a convex subset in* $C(T)$.

*Proof*

*f*and

*g*in ${B}_{0}$, we let

*v*belongs to ${B}_{0}$. Thus we have

This completes the proof of Theorem 3. □

Making use of Theorem 3 and (2.4), we can derive the following.

**Corollary 2**

*If*

*φ*

*is an extreme point of*$\mathcal{K}$,

*then*

*is a convex subset in* $C(T)$.

holds for every $f\in \mathcal{S}$ (see [[3], p.70, Exercise 6]).

In view of the inequality (2.5), we have a generalization of [[1], Theorem 2] as follows.

**Theorem 4**

*If*$f\in {B}_{0}$

*and*$\phi \in \mathcal{S}$,

*then the*

*nth*

*L*-

*derivative of*${F}_{\phi}$

*at*

*f*

*satisfies*

**Remark** The proof would run parallel to that of [[1], Theorem 2] because there are many similarities. But, as we have seen in equation (1.2), we find it to be different from the definition of the class ${S}_{C}$, which was given by Nikić [1]. So, we include the proof of Theorem 4.

*Proof*Applying (1.2) and (1.3), it is not difficult to show that

*n*th derivative of

*φ*. Since

*T*is a compact metric space, there exists a point $\xi \in T$ such that

Combining (2.6) and (2.7), we obtain the desired result. □

## 3 Examples

**Example 1**Let the function

*φ*be defined by (1.1). For a fixed radius $0<r<1$, we let $T=\{z\in \mathbb{C}:|z|\le r\}$. If we define a continuous function $f:T\to \mathbb{C}$ by $f(z)=z$, then

on *T*.

**Example 2**Setting $\phi (z)=z$ in (1.2), we have

**Example 3**If $\phi \in \mathcal{A}$ satisfies

*φ*is univalent. Hence, by Theorem 1, we obtain

## Declarations

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

## Authors’ Affiliations

## References

- Nikić M: Koebe’s and Bieberbach’s inequalities in the Banach algebra of continuous functions.
*J. Math. Anal. Appl.*1996, 199: 149–156. 10.1006/jmaa.1996.0132MathSciNetView ArticleGoogle Scholar - de Branges L: A proof of the Bieberbach conjecture.
*Acta Math.*1985, 154: 137–152. 10.1007/BF02392821MathSciNetView ArticleGoogle Scholar - Duren PL A Series of Comprehensive Studies in Mathematics 259. In
*Univalent Functions*. Springer, Berlin; 1983.Google Scholar

## Copyright

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