Advances in Pure Mathematics, 2012, 2, 200-202
http://dx.doi.org/10.4236/apm.2012.23028 Published Online May 2012 (http://www.SciRP.org/journal/apm)
An Integral Representation of a Family of Slit Mappings
Adrian W. Cartier, Michael P. Sterner
Department of Biology-Chemistry-Mathematics, University of Montevallo, Montevallo, USA
Received January 4, 2012; revised February 17, 2012; accepted February 28, 2012
ABSTRACT
We consider a normalized family F of analytic functions f, whose common domain is the complement of a closed ray in
the complex plane. If

f
z is real when z is real and the range of f does not intersect the nonpositive real axis, then f
can be reproduced by integrating the biquadratic kernel



2
2
11
1
z z
tz
tt


 against a probability measure


t  . It is
shown that while this integral representation does not characterize the family F, it applies to a large class of functions,
including a collection of functions which multiply the Hardy space Hp into itself.
Keywords: Herglotz Formula; Integral Representations; Subordination; Slit Mappings; Hardy Spaces; Multipliers;
Hadamard Product
1. Introduction
Let

:z

1zΔ, and let

:1.zCz 

Δ
Suppose f is analytic in Δ with the real part of f nonnega-
tive. Then there is a nondecreasing function μ defined on

0, 2π

such that

d
ez
2π
0
it
it
f
z


tib
ez


, where b
is a real constant. This representation of such functions
by integrating a bilinear kernel against a measure is due
to G. Herglotz ([1], pp. 21-24) and ([2], pp. 27-30). In
this paper, we examine a family of functions defined on
the complex plane with a closed ray removed, which may
be represented by integrating a biquadratic kernel against
a probability measure (A measure μ is called a probabil-
ity measure on


0, 1
11

provided μ is nonnegative with
0). In what follows, given functions f and g
analytic in Δ, we say that f is subordinate to g (written

dt 
f
g) provided


 
f
zg  z for some

analytic
in Δ with

.

zz  
2. The Main Results
Theorem 1. Let ,
 
1,C 

,0C




01f

, and let
F be the family of functions f having the following prop-
erties:
1) f is analytic in ;
2) ;
3)
f
z1z 

f
 

R whenever ;

4) .
Then

2
1
2
0
11
:d,
1
ttz z
F
ffz t
tz














where μ is a probability measure.
Proof. Let
2
11.
1
w
ww








Then

is an ana-
lytic, bijective mapping of Δ in the w-plane onto in
the z-plane with
Ω


00.


Let . Then fF


ΩΦf
, gf by 4). Let



and let

2
1.
1
w
w





Gw Then
G is an analytic, bijective mapping of Δ onto

with


s
g
.G Define G
.hG
to be the collection of all func-
tions h analytic in Δ with By a result due to D.
A. Brannan, J. G. Clunie, and W. E. Kirwan [3],
 
2
Δ
1
analytic inΔ:d
1
z
co sGhhzz


,


 






 






where v is a probability measure and

co sG

denotes
the closed convex hull of G

1.Fz z. Let
s


:ΩΦF
Then is an analytic bijection with


01.F

Since


g
sG

,

2
Δ
1d
1
w
gw w




 




Δw

and v a probability measure. Since for

is in-
jective with


ΔΩ,

we have









g
wf wfz  .
C
opyright © 2012 SciRes. APM

A. W. CARTIER, M. P. STERNER 201
Hence
 
 




2
1
d.
1
z
z

2
1
1
Δ
2
Δ
Δ
1d
1
11
111 d
11
111
11
11
z
fz z
z
z
z
z
 








































By 3)

f
zf

,1 .z

z whenever Since
is symmetric about the real axis, by the identity theo- Ω
rem

fz fz

:Im 0
throughout Ω. Let
.X

 

For any measurable subset A of
X define
 

*12 .
A
AA

  We have
 



 



 


 
2
22
2
22
0
2
π
2
1
2
0
1
2
11
1
211
111
111
Re1 4
Re 14Re 1
14 cos11
1cos
12
11
d.
1
X
fzfz fz
z
z
zz
zz
z
ttz zt
tz





2
*
1
1
d
4
d
4
d()
z
z
z


















 


 


 





























where

*i
e

  

and



1
cos21 .tt


 
:1,

This integral representation does not characterize F, as
the following theorem shows.
Theorem 2. Suppose
f
CC

is defined via
  
2
11
d
1
ttz z
fz t
tz




2
1
0
where

is a probability measure.
1) If

has support

, then

0,1 .fF
2) If

is a point mass, fF

if and only if

has
support


0 or

.
1
Proof. Let f be as defined in the theorem. Suppose

has support

, and the weight at 0 is a, where

0,1

1

0,
.
a Since

is a probability measure, the corre-
sponding weight at 1 is 1 – a. We have

221
.
1
az az
fz z


 Since 0, the value 1a

111za 
,fF

1
lies in the domain of f, and is mapped to
the origin in the w-plane. Therefore proving 1).
f
Observe that point mass at 0 gives zz 

and
point mass at 1 gives 1
1
fz z 

, each of which is an
analytic bijection from Ω onto , and clearly in F.
Suppose

has support


t01t

, where . Then



2
2
11
.
1
ttz z
fz
tz



 
Let

1141
.
21
tt
ttt

 



0t 


precisely when t = 1/2. It follows that Then


0, 1t, and

lies in the domain of f for each


0f 
f
F

. Therefore .

3. An Application
In [4], T. H. MacGregor and M. P. Sterner investigate
multipliers of Hardy spaces of analytic functions using
asymptotic expansions and power functions of the form

1b z



n
, where b is a complex constant. A subclass of F
which multiplies Hp into Hp is given in the following
theorem. Suppose 0n
n
f
zaz



n and
0n n
g
zbz



0
*
n
nn
n
 are analytic in Δ. Then the Hadamard
product of f and g is defined by
f
gz abz



Δ.z

We say that f multiplies Hp into Hp provided for
*p gHp whenever
f
g
H.
Theorem 3. Let

be a finite complex-valued Borel
measure defined on


0, 1

and let
 
1
0
1d.
1
fztz
tz
 



Then f is a multiplier of Hp into Hp for every p > 0.
Moreover, there is a constant Cp depending only on p
such that *
p
p
p
H
H
fgC g  .
p
g
H for all
Proof. Let f be as described in the hypotheses of the
theorem, and suppose
p
g
H
Δz
for some p > 0. Then for


and

0,1

r we have





2π
0
2π1
00
12π
00
1
*d
2π
11
dd
2π1
1dd .
2π1
ii
i
i
i
i
fgrzfzegre
tgre
tze
gre t
tze
























A. W. CARTIER, M. P. STERNER
202


1,C,


f
 the value of f is unity at the origin, and

By Cauchy’s formula,
 

 2π
0
1
2π
1d.
2π1
r
i
i
g
gz z
iz
gre
ze
r














d,
01 r r

*d.
z
is real when z is real


1z.

 

Finally, observe that
the range of f is contained in

,0 .C


1, .zC
To see this
last statement, fix  :0 1tz t Then




Hence

1
0
f
grzgrtz t 
Therefore for 01

 and 02π

 we have





d.
ii
1
0
*
f
gegte t


Let


01
x
 for sup
 i Ggxe

02π.

 Then
G is the Hardy-Littlewood maximal function for g, and
so lies in


0, 2π
p
L ([5], p. 12). Moreover, there is a
constant Cp depending only on p such that
p
p
p
L
H
GCg (In fact, for , ). Since 11
p
Cp
01

 and 0, we obtain 1t

 
 
1
001
*sup
ii
x
fg egxet

d .G

 


Hence



2π2π
00
11
*d
2π2π
p
i
p
p
fg e

d
.
p
p
pp
H
G
Cg







Therefore


1
2π
0
01
1
sup* d
2π.
p
p
p
i
fge








pH
C g 
If we restrict the measure

to be a probability meas-
ure, then the formula implies the analyticity of f on
is the line segment from 0 to z. Hence 1:01
1t tz






is the arc of the circle determined by 1, 1
1z, and 0,
having endpoints 1 and 1
1z

and not including the ori-
gin. Since

is a probability measure,

1
0
1d
1t tz



lies in the circular segment which is the closed convex
hull of that arc, and this circular segment does not inter-
sect


,0 . Hence each such multiplier function f lies
in F.
REFERENCES
[1] P. L. Duren, “Univalent Functions,” Springer-Verlag, New
York, 1983.
[2] D. J. Hallenbeck and T. H. MacGregor, “Linear Problems
and Convexity Techniques in Geometric Function The-
ory,” Pitman Publishing Ltd., London, 1984.
[3] D. A. Brannan, J. G. Clunie and W. E. Kirwan, “On the
Coefficient Problem for Functions of Bounded Boundary
Rotation,” Annales Academiae Scientiarum Fennicae. Se-
ries AI. Mathematica, Vol. 523, 1972, pp. 403-489.
[4] T. H. MacGregor and M. P. Sterner, “Hadamard Products
with Power Functions and Multipliers of Hardy Spaces,”
Journal of Mathematical Analysis and Applications, Vol.
282, No. 1, 2003, pp. 163-176.
doi:10.1016/S0022-247X(03)00128-8
[5] P. L. Duren, “Theory of Hp Spaces,” Academic Press, New
York, 1970.