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# Univalent functions in the Banach algebra of continuous functions

*Journal of Inequalities and Applications*
**volume 2013**, Article number: 145 (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.

## 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).

Let \mathcal{A} denote the class of functions \phi (z) of the form

which are analytic in the open unit disk

Also, let \mathcal{S} denote the class of all functions in \mathcal{A} which are *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}.

Corresponding to the function \phi \in \mathcal{A}, we define a composition operator {F}_{\phi}:{B}_{0}\to C(T) by

We denote by {S}_{C} the class of all functions {F}_{\phi} which are injective in the open unit ball {B}_{0}. We note that Nikić ([1], Definition 2) defined a similar class {S}_{C} without using the function *φ*. In this case, we cannot ensure the convergence of the series

Now we let *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

and such that

for all h\in B(0:r). We call *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.

(⇒) Let \phi ({z}_{1})=\phi ({z}_{2}) for {z}_{1} and {z}_{2} in \mathcal{U}. If we take the constant functions *f* and *g* such that f={z}_{1} and g={z}_{2}, then it is obvious that

Furthermore, from (1.2) it is easy to see 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

By the hypothesis,

for all f\in {B}_{0}. It follows from (1.3) that

Since

and

equations (2.1) and (2.3) imply that

Hence

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*

Assume that

For the functions *f* and *g* in {B}_{0}, we let

and

Then we have

Since

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

This completes the proof of Theorem 3. □

We now recall that the function

is the well-known extremal function (see [3]) for the class \mathcal{K} of convex functions. If we let

then we note that

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).

It is well known that the sharp inequality

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

where {\phi}^{(n)} is the *n* th derivative of *φ*. Since

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

Since \phi \in \mathcal{S}, from (2.4) we have

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

then the Noshiro-Warschawski theorem implies that *φ* is univalent. Hence, by Theorem 1, we obtain

## 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.1996.0132de Branges L: A proof of the Bieberbach conjecture.

*Acta Math.*1985, 154: 137–152. 10.1007/BF02392821Duren PL A Series of Comprehensive Studies in Mathematics 259. In

*Univalent Functions*. Springer, Berlin; 1983.

## Acknowledgements

Dedicated to Professor Hari M Srivastava.

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Kim, Y.C., Choi, J.H. Univalent functions in the Banach algebra of continuous functions.
*J Inequal Appl* **2013**, 145 (2013). https://doi.org/10.1186/1029-242X-2013-145

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

### Keywords

- analytic function
- univalent function
- Banach algebra
- Noshiro-Warschawski theorem