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: sternerm@montevallo.edu

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

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

1zΔ, 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

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

g) provided

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)

z1z

f

R

whenever ;

4) .

Then

2

1

2

0

11

:d,

1

ttz z

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

.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

:ΩΦF

Then is an analytic bijection with

01.F

Since

sG

,

2

Δ

1d

1

w

gw w

Δw

and v a probability measure. Since for

is in-

jective with

ΔΩ,

we have

wf wfz

.

C

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