Open Access

Univalent functions in the Banach algebra of continuous functions

Journal of Inequalities and Applications20132013:145

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

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

analytic function univalent function Banach algebra Noshiro-Warschawski theorem

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 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 A denote the class of functions φ ( z ) of the form
φ ( z ) = z + n = 2 a n z n ,
(1.1)
which are analytic in the open unit disk
U = { z : z C  and  | z | < 1 } .
Also, let S denote the class of all functions in A which are univalent in the unit disk U . A function φ ( z ) belonging to the class S is said to be convex in U if and only if
{ 1 + z φ ( z ) φ ( z ) } > 0 ( z U ) .

We denote by K the class of all functions in S which are convex in U .

Corresponding to the function φ A , we define a composition operator F φ : B 0 C ( T ) by
F φ ( f ) = φ f = f + n = 2 a n f n .
(1.2)
We denote by S C the class of all functions F φ 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 + n = 2 a n f n .
Now we let G be an open nonempty subset of C ( T ) . A function F : G C ( T ) is said to be L-differentiable at a point f G if there exists λ C ( T ) and a map η defined in a ball B ( 0 : r ) with values in C ( T ) such that
lim h 0 η ( h ) h = 0
and such that
F ( f + h ) F ( f ) = λ h + η ( h )
for all h B ( 0 : r ) . We call λ the L-derivative of F at f and denote it by F ( f ) . From [1], we see that
F φ ( f ) = φ f ,
(1.3)

where φ 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 φ

We begin by proving the following theorem.

Theorem 1 F φ S C if and only if φ S .

Proof () Suppose that F φ ( f ) = F φ ( g ) for the functions f and g in B 0 . Then it means that
φ ( f ( t ) ) = φ ( g ( t ) )

for all t T . Since φ is univalent, f ( t ) = g ( t ) for all t T .

() Let φ ( z 1 ) = φ ( z 2 ) for z 1 and z 2 in 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 B 0 and g B 0 .
Furthermore, from (1.2) it is easy to see that
F φ ( f ) = F φ ( g ) .

Since F φ 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 φ ( f ) = f + n = 2 a n f n S C ,
then
| a n | 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 φ ( f ) has a positive real part for all f B 0 , then
F φ S C .
Proof If f 1 B 0 , f 2 B 0 and f 1 f 2 , then there exists t T such that
f 1 ( t ) f 2 ( t ) .
(2.1)
By the hypothesis,
{ F φ ( f ) } > 0
(2.2)
for all f B 0 . It follows from (1.3) that
{ φ ( f ( t ) ) } > 0 ( f B 0 : t T ) .
(2.3)
Since
φ ( f 2 ( t ) ) φ ( f 1 ( t ) ) = f 1 ( t ) f 2 ( t ) φ ( x ) d x = ( f 2 ( t ) f 1 ( t ) ) 0 1 φ ( λ f 2 ( t ) + ( 1 λ ) f 1 ( t ) ) d λ
and
λ f 2 ( t ) + ( 1 λ ) f 1 ( t ) B 0 ,
equations (2.1) and (2.3) imply that
φ ( f 2 ( t ) ) φ ( f 1 ( t ) ) .
Hence
F φ ( f 1 ( t ) ) F φ ( f 2 ( t ) )

at t T , which shows that F φ is injective. □

Remark Since T is compact, { f ( t ) : t T } is a closed proper subset of U . Hence the condition (2.2) does not imply
{ φ ( z ) } > 0 ( z U ) .

Next we obtain the following.

Theorem 3 Let
φ ( z ) = z 1 z .
Then
{ F φ ( f ) : f B 0 }

is a convex subset in C ( T ) .

Proof

Assume that
α > 0 , β > 0 and α + β = 1 .
For the functions f and g in B 0 , we let
u ( t ) α F φ ( f ( t ) ) + β F φ ( g ( t ) )
and
v ( t ) u ( t ) 1 + u ( t ) .
Then we have
u ( t ) = v ( t ) 1 v ( t ) = F φ ( v ( t ) ) .
Since
1 | v ( t ) | 2 = 1 v ( t ) v ( t ) ¯ = 1 u ( t ) 1 + u ( t ) u ( t ) ¯ 1 + u ( t ) ¯ = 1 1 + u ( t ) ¯ ( 1 + u ( t ) + u ( t ) ¯ ) 1 1 + u ( t ) = 1 + 2 { u ( t ) } 1 + | u ( t ) | 2 > 0 ,
the function v belongs to B 0 . Thus we have
u = F φ ( v ) { F φ ( f ) : f B 0 } .

This completes the proof of Theorem 3. □

We now recall that the function
φ η ( z ) = z 1 η z ( η C , | η | = 1 )
is the well-known extremal function (see [3]) for the class K of convex functions. If we let
φ ( z ) = z 1 z ,
then we note that
φ η ( z ) = η 1 φ ( η 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 K , then
{ F φ ( f ) : f B 0 }

is a convex subset in C ( T ) .

It is well known that the sharp inequality
| f ( n ) ( z ) | n ! ( n + | z | ) ( 1 | z | ) n + 2 ( n = 1 , 2 , 3 , )
(2.5)

holds for every f 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 B 0 and φ S , then the nth L-derivative of F φ at f satisfies
F ( n ) ( f ) n ! ( n + f ) ( 1 f ) 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 φ ( n ) ( f ) = φ ( n ) f ( n = 1 , 2 , 3 , ) ,
where φ ( n ) is the n th derivative of φ. Since
F φ ( n ) ( f ) C ( T )
and T is a compact metric space, there exists a point ξ T such that
F φ ( n ) ( f ) = | F φ ( n ) ( f ( ξ ) ) | = | φ ( n ) ( f ( ξ ) ) | .
(2.6)
Since φ S , from (2.4) we have
| φ ( n ) ( f ( ξ ) ) | n ! ( n + | f ( ξ ) | ) ( 1 | f ( ξ ) | ) n + 2 n ! ( n + f ) ( 1 f ) 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 C : | z | r } . If we define a continuous function f : T C by f ( z ) = z , then
F φ ( f ) = φ

on T.

Example 2 Setting φ ( z ) = z in (1.2), we have
F φ ( f ) = f .
Example 3 If φ A satisfies
{ φ ( z ) } > 0 ( z U ) ,
then the Noshiro-Warschawski theorem implies that φ is univalent. Hence, by Theorem 1, we obtain
F φ S C .

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

Authors’ Affiliations

(1)
Department of Mathematics Education, Yeungnam University
(2)
Department of Mathematics Education, Daegu National University of Education

References

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  2. de Branges L: A proof of the Bieberbach conjecture. Acta Math. 1985, 154: 137–152. 10.1007/BF02392821MathSciNetView ArticleGoogle Scholar
  3. Duren PL A Series of Comprehensive Studies in Mathematics 259. In Univalent Functions. Springer, Berlin; 1983.Google Scholar

Copyright

© Kim and Choi; licensee Springer 2013

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