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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 fC(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(zU).

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

Corresponding to the function φA, we define a composition operator F φ : B 0 C(T) by

F φ (f)=φf=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:GC(T) is said to be L-differentiable at a point fG if there exists λC(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)=λh+η(h)

for all hB(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 φS.

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

() 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 andg 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 tT 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 :tT).
(2.3)

Since

φ ( f 2 ( t ) ) φ ( f 1 ( t ) ) = f 1 ( t ) f 2 ( t ) φ (x)dx= ( f 2 ( t ) f 1 ( t ) ) 0 1 φ ( λ f 2 ( t ) + ( 1 λ ) f 1 ( t ) ) dλ

and

λ f 2 (t)+(1λ) 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 tT, which shows that F φ is injective. □

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

{ φ ( z ) } >0(zU).

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>0andα+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(ηz).
(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 fS (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 φS, 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 ξT such that

F φ ( n ) ( f ) = | F φ ( n ) ( f ( ξ ) ) | = | φ ( n ) ( f ( ξ ) ) | .
(2.6)

Since φS, 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={zC:|z|r}. If we define a continuous function f:TC 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 φA satisfies

{ φ ( z ) } >0(zU),

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

F φ S C .

References

  1. 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

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  2. de Branges L: A proof of the Bieberbach conjecture. Acta Math. 1985, 154: 137–152. 10.1007/BF02392821

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  3. Duren PL A Series of Comprehensive Studies in Mathematics 259. In Univalent Functions. Springer, Berlin; 1983.

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Acknowledgements

Dedicated to Professor Hari M Srivastava.

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Correspondence to Jae Ho Choi.

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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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Keywords

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