An Integral Representation of a Family of Slit Mappings

doi:10.4236/apm.2012.23028

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

Email: [email protected]

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

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

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











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

A. W. CARTIER, M. P. STERNER 201

Hence

 







2

1

d.

1

z



2

1

Δ

2

Δ

1d

1

11

111 d

11

111

11

z

fz 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

Re1 4

Re 14Re 1

14 cos11

1cos

12

11

d.

1

X

fzfz fz

z

zz

z

ttz zt

tz





2

*

1

d

4

d

4

d()

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

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

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

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

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

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

fg e



d

.

p

pp

H

G

Cg













Therefore



1

2π

0

01

1

sup* d

2π.

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.

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