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Univalent functions in the Banach algebra of continuous functions
Journal of Inequalities and Applications volume 2013, Article number: 145 (2013)
In this paper, we investigate several interesting properties of a composition operator defined on the open unit ball of the Banach algebra . We also consider the Noshiro-Warschawski theorem in the Banach algebra of continuous functions.
1 Introduction and definitions
Throughout this paper, denotes the Banach algebra, with sup norm, of continuous complex-valued functions defined on a compact metric space T. Let be an open ball in centered at with radius r. In particular, for the sake of brevity, we use the simplified notation instead of .
Let denote the class of functions of the form
which are analytic in the open unit disk
Also, let denote the class of all functions in which are univalent in the unit disk . A function belonging to the class is said to be convex in if and only if
We denote by the class of all functions in which are convex in .
Corresponding to the function , we define a composition operator by
We denote by the class of all functions which are injective in the open unit ball . We note that Nikić (, Definition 2) defined a similar class 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 . A function is said to be L-differentiable at a point if there exists and a map η defined in a ball with values in such that
and such that
for all . We call λ the L-derivative of F at f and denote it by . From , we see that
where is a derivative of φ.
In the present paper, we investigate several geometric properties of the class associated with the theory of univalent functions.
2 Geometric properties of the composition operator
We begin by proving the following theorem.
Theorem 1 if and only if .
Proof (⇐) Suppose that for the functions f and g in . Then it means that
for all . Since φ is univalent, for all .
(⇒) Let for and in . If we take the constant functions f and g such that and , then it is obvious that
Furthermore, from (1.2) it is easy to see that
Since is injective, we have . Hence we get . This completes the proof of Theorem 1. □
By using Brange’s theorem , we obtain the following.
Corollary 1 If
Now we prove the Noshiro-Warschawski theorem (, Theorem 2.16) in the Banach algebra .
Theorem 2 If the L-derivative has a positive real part for all , then
Proof If , and , then there exists such that
By the hypothesis,
for all . It follows from (1.3) that
equations (2.1) and (2.3) imply that
at , which shows that is injective. □
Remark Since T is compact, is a closed proper subset of . Hence the condition (2.2) does not imply
Next we obtain the following.
Theorem 3 Let
is a convex subset in .
For the functions f and g in , we let
Then we have
the function v belongs to . Thus we have
This completes the proof of Theorem 3. □
We now recall that the function
is the well-known extremal function (see ) for the class 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 , then
is a convex subset in .
It is well known that the sharp inequality
holds for every (see [, p.70, Exercise 6]).
In view of the inequality (2.5), we have a generalization of [, Theorem 2] as follows.
Theorem 4 If and , then the nth L-derivative of at f satisfies
Remark The proof would run parallel to that of [, 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 , which was given by Nikić . So, we include the proof of Theorem 4.
Proof Applying (1.2) and (1.3), it is not difficult to show that
where is the n th derivative of φ. Since
and T is a compact metric space, there exists a point such that
Since , from (2.4) we have
Combining (2.6) and (2.7), we obtain the desired result. □
Example 1 Let the function φ be defined by (1.1). For a fixed radius , we let . If we define a continuous function by , then
Example 2 Setting in (1.2), we have
Example 3 If satisfies
then the Noshiro-Warschawski theorem implies that φ is univalent. Hence, by Theorem 1, we obtain
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.1996.0132
de Branges L: A proof of the Bieberbach conjecture. Acta Math. 1985, 154: 137–152. 10.1007/BF02392821
Duren PL A Series of Comprehensive Studies in Mathematics 259. In Univalent Functions. Springer, Berlin; 1983.
Dedicated to Professor Hari M Srivastava.
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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Cite this article
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
- analytic function
- univalent function
- Banach algebra
- Noshiro-Warschawski theorem