An Integral Representation of a Family of Slit Mappings

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 1 1 1 z z tz t t     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 H into itself.


Introduction
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 probability measure on   0,1 . In what follows, given functions f and g analytic in Δ, we say that f is subordinate to g (written

The Main Results
Theorem 1.Let , , and let F be the family of functions f having the following properties: where μ is a probability measure.
Then  is an ana- lytic, bijective mapping of Δ in the w-plane onto in the z-plane with to be the collection of all functions h analytic in Δ with By a result due to D. A. Brannan, J. G. Clunie, and W. E. Kirwan [3], where v is a probability measure and Since is symmetric about the real axis, by the identity theo-

 
This integral representation does not characterize F, as the following theorem shows.
where  is a probability measure.
Proof.Let f be as defined in the theorem.Suppose  has support  , and the weight at 0 is a, where  Since  is a probability measure, the corre- sponding weight at 1 is 1 -a.We have 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 z z      and point mass at 1 gives , each of which is an analytic bijection from Ω onto , and clearly in F.
, and  lies in the domain of f for each

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 , where b is a complex constant.A subclass of F which multiplies H p into H p is given in the following theorem.Suppose Theorem 3. Let  be a finite complex-valued Borel measure defined on   p g H  for all Proof.Let f be as described in the hypotheses of the theorem, and suppose f  the value of f is unity at the origin, and  By Cauchy's formula,


Hence each such multiplier function f lies in F.
Suppose f is analytic in Δ with the real part of f nonnegative.Then there is a nondecreasing function μ defined on 


are analytic in Δ.Then the Hadamard product of f and g is defined by


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 circular segment which is the closed convex hull of that arc, and this circular segment does not intersect   ,0 .
is the Hardy-Littlewood maximal function for g, and so lies in