R E S E A R C H
Open Access
Univalent functions in the Banach algebra of
continuous functions
Yong Chan Kim
1and Jae Ho Choi
2**Correspondence:
2Department of Mathematics
Education, Daegu National University of Education, Daegu, 705-715, Korea
Full list of author information is available at the end of the article
Abstract
In this paper, we investigate several interesting properties of a composition operator defined on the open unit ballB0of the Banach algebraC(T). We also consider the Noshiro-Warschawski theorem in the Banach algebra of continuous functions.
MSC: Primary 30C45; secondary 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 spaceT. LetB(f :r) be an open ball inC(T) centered atf ∈C(T) with radiusr. In particular, for the sake of brevity, we use the simplified notationBinstead ofB( : ).
LetAdenote the class of functionsϕ(z) of the form
ϕ(z) =z+
∞
n=
anzn, (.)
which are analytic in the open unit disk
U=z:z∈Cand|z|< .
Also, letS denote the class of all functions inAwhich areunivalentin the unit diskU. A functionϕ(z) belonging to the classSis said to beconvexinUif and only if
+zϕ
(z)
ϕ(z)
> (z∈U).
We denote byKthe class of all functions inSwhich are convex inU.
Corresponding to the functionϕ∈A, we define a composition operatorFϕ:B→C(T)
by
Fϕ(f) =ϕ◦f =f+
∞
n=
anfn. (.)
We denote bySCthe class of all functionsFϕwhich are injective in the open unit ballB. We
note that Nikić ([], Definition ) defined a similar classSCwithout using the functionϕ.
In this case, we cannot ensure the convergence of the series
f +
∞
n=
anfn.
Now we letGbe an open nonempty subset ofC(T). A functionF:G→C(T) is said to beL-differentiableat a pointf ∈Gif there existsλ∈C(T) and a mapηdefined in a ball
B( :r) with values inC(T) such that
lim
h→
η(h)
h =
and such that
F(f+h) –F(f) =λh+η(h)
for allh∈B( :r). We callλtheL-derivativeofFatf and denote it byF(f). From [], we see that
Fϕ(f) =ϕ◦f, (.)
whereϕis a derivative ofϕ.
In the present paper, we investigate several geometric properties of the classSC
associ-ated with the theory of univalent functions.
2 Geometric properties of the composition operatorFϕ We begin by proving the following theorem.
Theorem Fϕ∈SCif and only ifϕ∈S.
Proof (⇐) Suppose thatFϕ(f) =Fϕ(g) for the functionsf andginB. Then it means that
ϕf(t)=ϕg(t)
for allt∈T. Sinceϕis univalent,f(t) =g(t) for allt∈T.
(⇒) Letϕ(z) =ϕ(z) forzandzinU. If we take the constant functionsf andgsuch
thatf =zandg=z, then it is obvious that
f ∈B and g∈B.
Furthermore, from (.) it is easy to see that
Fϕ(f) =Fϕ(g).
SinceFϕ is injective, we havef =g. Hence we getz=z. This completes the proof of
By using Brange’s theorem [], we obtain the following.
Corollary If
Fϕ(f) =f +
∞
n=
anfn∈SC,
then
|an| ≤n.
Now we prove the Noshiro-Warschawski theorem ([], Theorem .) in the Banach algebraC(T).
Theorem If the L-derivative Fϕ(f)has a positive real part for all f ∈B,then
Fϕ∈SC.
Proof Iff∈B,f∈Bandf=f, then there existst∈Tsuch that
f(t)=f(t). (.)
By the hypothesis,
Fϕ(f)
> (.)
for allf ∈B. It follows from (.) that
ϕf(t)> (f ∈B:t∈T). (.)
Since
ϕf(t)
–ϕf(t)
=
f(t)
f(t)
ϕ(x)dx=f(t) –f(t)
ϕλf(t) + ( –λ)f(t)
dλ
and
λf(t) + ( –λ)f(t)∈B,
equations (.) and (.) imply that
ϕf(t)
=ϕf(t)
.
Hence
Fϕ
f(t)
=Fϕ
f(t)
Remark SinceTis compact,{f(t) :t∈T}is a closed proper subset ofU. Hence the con-dition (.) does not imply
ϕ(z)> (z∈U).
Next we obtain the following.
Theorem Let
ϕ(z) = z –z. Then
Fϕ(f) :f∈B
is a convex subset in C(T).
Proof Assume that
α> , β> and α+β= .
For the functionsf andginB, we let
u(t)≡αFϕ
f(t)+βFϕ
g(t) and
v(t)≡ u(t) +u(t). Then we have
u(t) = v(t) –v(t)=Fϕ
v(t).
Since
–v(t)= –v(t)v(t) = – u(t)
+u(t)
u(t) +u(t) =
+u(t)
+u(t) +u(t) +u(t) = + {u(t)}
+|u(t)| > ,
the functionvbelongs toB. Thus we have
u=Fϕ(v)∈
Fϕ(f) :f ∈B
.
We now recall that the function
ϕη(z) =
z
–ηz
η∈C,|η|=
is the well-known extremal function (see []) for the classKof convex functions. If we let
ϕ(z) = z –z,
then we note that
ϕη(z) =η–ϕ(ηz). (.)
Making use of Theorem and (.), we can derive the following.
Corollary Ifϕis an extreme point ofK,then
Fϕ(f) :f∈B
is a convex subset in C(T).
It is well known that the sharp inequality
f(n)(z)≤ n!(n+|z|)
( –|z|)n+ (n= , , , . . .) (.)
holds for everyf ∈S(see [, p., Exercise ]).
In view of the inequality (.), we have a generalization of [, Theorem ] as follows.
Theorem If f ∈Bandϕ∈S,then the nth L-derivative of Fϕat f satisfies
F(n)(f)≤ n!(n+f) ( –f)n+.
Remark The proof would run parallel to that of [, Theorem ] because there are many similarities. But, as we have seen in equation (.), we find it to be different from the defi-nition of the classSC, which was given by Nikić []. So, we include the proof of Theorem .
Proof Applying (.) and (.), it is not difficult to show that
Fϕ(n)(f) =ϕ(n)◦f (n= , , , . . .), whereϕ(n)is thenth derivative ofϕ. Since
Fϕ(n)(f)∈C(T)
andTis a compact metric space, there exists a pointξ∈Tsuch that Fϕ(n)(f)=Fϕ(n)
Sinceϕ∈S, from (.) we have
ϕ(n)f(ξ)≤ n!(n+|f(ξ)|) ( –|f(ξ)|)n+ ≤
n!(n+f)
( –f)n+. (.)
Combining (.) and (.), we obtain the desired result.
3 Examples
Example Let the functionϕbe defined by (.). For a fixed radius <r< , we letT=
{z∈C:|z| ≤r}. If we define a continuous functionf :T→Cbyf(z) =z, then
Fϕ(f) =ϕ
onT.
Example Settingϕ(z) =zin (.), we have
Fϕ(f) =f.
Example Ifϕ∈Asatisfies
ϕ(z)> (z∈U),
then the Noshiro-Warschawski theorem implies thatϕis univalent. Hence, by Theorem , we obtain
Fϕ∈SC.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Author details
1Department of Mathematics Education, Yeungnam University, Gyongsan, 712-749, Korea.2Department of Mathematics
Education, Daegu National University of Education, Daegu, 705-715, Korea.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
Received: 24 December 2012 Accepted: 7 March 2013 Published: 2 April 2013
References
1. Niki´c, M: Koebe’s and Bieberbach’s inequalities in the Banach algebra of continuous functions. J. Math. Anal. Appl. 199, 149-156 (1996)
2. de Branges, L: A proof of the Bieberbach conjecture. Acta Math.154, 137-152 (1985)
3. Duren, PL: Univalent Functions. A Series of Comprehensive Studies in Mathematics, vol. 259. Springer, Berlin (1983)
doi:10.1186/1029-242X-2013-145
