Univalent functions in the Banach algebra of continuous functions
© Kim and Choi; licensee Springer 2013
Received: 24 December 2012
Accepted: 7 March 2013
Published: 2 April 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 .
We denote by the class of all functions in which are convex in .
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 .
for all . Since φ is univalent, for all .
Since is injective, we have . Hence we get . This completes the proof of Theorem 1. □
By using Brange’s theorem , we obtain the following.
Now we prove the Noshiro-Warschawski theorem (, Theorem 2.16) in the Banach algebra .
at , which shows that is injective. □
Next we obtain the following.
is a convex subset in .
This completes the proof of Theorem 3. □
Making use of Theorem 3 and (2.4), we can derive the following.
is a convex subset in .
holds for every (see [, p.70, Exercise 6]).
In view of the inequality (2.5), we have a generalization of [, Theorem 2] as follows.
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.
Combining (2.6) and (2.7), we obtain the desired result. □
Dedicated to Professor Hari M Srivastava.
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