Univalent functions in the Banach algebra of continuous functions

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

$$\phi (z)=z+\sum _{n=2}^{\mathrm{\infty }}{a}_n{z}^n,$$

(1.1)

which are analytic in the open unit disk

$$\mathcal{U}=\{z:z\in \mathbb{C}\text{ and }|z|<1\}.$$

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

$$\mathrm{\Re }\{1+\frac{z{\phi }^{″}(z)}{{\phi }^{\prime }(z)}\}>0(z\in \mathcal{U}).$$

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

$$F_{\phi }(f)=\phi \circ f=f+\sum _{n=2}^{\mathrm{\infty }}{a}_n{f}^n.$$

(1.2)

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

$$f+\sum _{n=2}^{\mathrm{\infty }}{a}_n{f}^n.$$

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

$$\underset{h\to 0}{lim}\frac{\eta (h)}{\parallel h\parallel }=0$$

and such that

$$F(f+h)-F(f)=\lambda h+\eta (h)$$

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

$$F_{\phi }^{\mathrm{\prime }}(f)={\phi }^{\mathrm{\prime }}\circ f,$$

(1.3)

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

$$\phi (f(t))=\phi (g(t))$$

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

$$f\in B_0\text{and}g\in B_0.$$

Furthermore, from (1.2) it is easy to see that

$$F_{\phi }(f)=F_{\phi }(g).$$

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

$$F_{\phi }(f)=f+\sum _{n=2}^{\mathrm{\infty }}{a}_n{f}^n\in S_C,$$

then

$$|a_n|\le n.$$

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

$$F_{\phi }\in S_C.$$

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

$$f_1(t)\ne f_2(t).$$

(2.1)

By the hypothesis,

$$\mathrm{\Re }\{F_{\phi }^{\mathrm{\prime }}(f)\}>0$$

(2.2)

for all $f\in B_0$. It follows from (1.3) that

$$\mathrm{\Re }\left\{{\phi }^{\mathrm{\prime }}(f(t))\right\}>0(f\in B_0:t\in T).$$

(2.3)

Since

$$\phi (f_2(t))-\phi (f_1(t))={\int }_{f_1(t)}^{f_2(t)}{\phi }^{\mathrm{\prime }}(x)dx=(f_2(t)-f_1(t)){\int }_0^1{\phi }^{\mathrm{\prime }}(\lambda f_2(t)+(1-\lambda )f_1(t))d\lambda$$

and

$$\lambda f_2(t)+(1-\lambda )f_1(t)\in B_0,$$

equations (2.1) and (2.3) imply that

$$\phi (f_2(t))\ne \phi (f_1(t)).$$

Hence

$$F_{\phi }(f_1(t))\ne F_{\phi }(f_2(t))$$

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

$$\mathrm{\Re }\{{\phi }^{\mathrm{\prime }}(z)\}>0(z\in \mathcal{U}).$$

Next we obtain the following.

Theorem 3 Let

$$\phi (z)=\frac{z}{1-z}.$$

Then

$$\{F_{\phi }(f):f\in B_0\}$$

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

Proof

Assume that

$$\alpha >0,\beta >0\text{and}\alpha +\beta =1.$$

For the functions f and g in $B_0$, we let

$$u(t)\equiv \alpha F_{\phi }(f(t))+\beta F_{\phi }(g(t))$$

and

$$v(t)\equiv \frac{u(t)}{1+u(t)}.$$

Then we have

$$u(t)=\frac{v(t)}{1-v(t)}=F_{\phi }(v(t)).$$

Since

$$\begin{array}{rcl}1-{|v(t)|}^2& =& 1-v(t)\overline{v(t)}\\ =& 1-\frac{u(t)}{1+u(t)}\frac{\overline{u(t)}}{1+\overline{u(t)}}\\ =& \frac{1}{1+\overline{u(t)}}(1+u(t)+\overline{u(t)})\frac{1}{1+u(t)}\\ =& \frac{1+2\mathrm{\Re }\{u(t)\}}{1+{|u(t)|}^2}>0,\end{array}$$

the function v belongs to $B_0$. Thus we have

$$u=F_{\phi }(v)\in \{F_{\phi }(f):f\in B_0\}.$$

This completes the proof of Theorem 3. □

We now recall that the function

$${\phi }_{\eta }(z)=\frac{z}{1-\eta z}(\eta \in \mathbb{C},|\eta |=1)$$

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

$$\phi (z)=\frac{z}{1-z},$$

then we note that

$${\phi }_{\eta }(z)={\eta }^{-1}\phi (\eta z).$$

(2.4)

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

$$\{F_{\phi }(f):f\in B_0\}$$

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

It is well known that the sharp inequality

$$|f^{(n)}(z)|\le \frac{n!(n+|z|)}{{(1-|z|)}^{n+2}}(n=1,2,3,\dots )$$

(2.5)

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

$$\parallel F^{(n)}(f)\parallel \le \frac{n!(n+\parallel f\parallel )}{{(1-\parallel f\parallel )}^{n+2}}.$$

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

$$F_{\phi }^{(n)}(f)={\phi }^{(n)}\circ f(n=1,2,3,\dots ),$$

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

$$F_{\phi }^{(n)}(f)\in C(T)$$

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

$$\parallel F_{\phi }^{(n)}(f)\parallel =\left|F_{\phi }^{(n)}(f(\xi ))\right|=\left|{\phi }^{(n)}(f(\xi ))\right|.$$

(2.6)

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

$$\left|{\phi }^{(n)}(f(\xi ))\right|\le \frac{n!(n+|f(\xi )|)}{{(1-|f(\xi )|)}^{n+2}}\le \frac{n!(n+\parallel f\parallel )}{{(1-\parallel f\parallel )}^{n+2}}.$$

(2.7)

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

$$F_{\phi }(f)=\phi$$

on T.

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

$$F_{\phi }(f)=f.$$

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

$$\mathrm{\Re }\{{\phi }^{\mathrm{\prime }}(z)\}>0(z\in \mathcal{U}),$$

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

$$F_{\phi }\in S_C.$$