 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 fz 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 22111z ztztt 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 dez2π0ititfztibez, where b is a real constant. This representation of such functions by integrating a bilinear kernel against a measure is due to G. Herglotz (, pp. 21-24) and (, 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, 111 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fg) provided fzg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)fz1z f R whenever ; 4) . Then 212011:d,1ttz zFffz ttz where μ is a probability measure. Proof. Let 211.1www 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 21.1wwGw Then G is an analytic, bijective mapping of Δ onto  with sg.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 ,  2Δ1analytic inΔ:d1zco 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 gsG, 2Δ1d1wgw wΔw  and v a probability measure. Since for  is in- jective with ΔΩ, we have gwf wfz. Copyright © 2012 SciRes. APM A. W. CARTIER, M. P. STERNER 201Hence   21d.1zz211Δ2ΔΔ1d111111 d111111111zfz zzzzz  By 3) fzf,1 .zz whenever Since is symmetric about the real axis, by the identity theo- Ωrem fz fz:Im 0 throughout Ω. Let .X  For any measurable subset A of X define *12 .AAA We have     22222202π212012111211111111Re1 4Re 14Re 114 cos111cos1211d.1Xfzfz fzzzzzzzzttz zttz2*11d4d4d()zzz    where *ie and 1cos21 .tt:1, This integral representation does not characterize F, as the following theorem shows. Theorem 2. Suppose fCC is defined via   211d1ttz zfz ttz210 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 . 1Proof. 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 221.1az azfz 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). fObserve that point mass at 0 gives zz  and point mass at 1 gives 11fz z, each of which is an analytic bijection from Ω onto , and clearly in F. Suppose  has support t01t, where . Then 2211.1ttz zfztz  Let 1141.21ttttt  0t precisely when t = 1/2. It follows that Then 0, 1t, and  lies in the domain of f for each 0ffF. Therefore . 3. An Application In , T. H. MacGregor and M. P. Sterner investigate multipliers of Hardy spaces of analytic functions using asymptotic expansions and power functions of the form 1bzn, where b is a complex constant. A subclass of F which multiplies Hp into Hp is given in the following theorem. Suppose 0nnfzazn and 0nngzbz0* nnnn are analytic in Δ. Then the Hadamard product of f and g is defined by fgz abzΔ.z  We say that f multiplies Hp into Hp provided for *pgHp whenever fgH. Theorem 3. Let  be a finite complex-valued Borel measure defined on 0, 1 and let  101d.1fztztz 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 *pppHHfgC g.pgH for all Proof. Let f be as described in the hypotheses of the theorem, and suppose pgHΔz for some p > 0. Then for  and 0,1r we have 2π02π10012π001*d 2π11dd2π11dd .2π1iiiiiifgrzfzegretgretzegre ttze Copyright © 2012 SciRes. APM A. W. CARTIER, M. P. STERNER Copyright © 2012 SciRes. APM 202 1,C,f the value of f is unity at the origin, and By Cauchy’s formula,  2π012π1d.2π1riiggz zizgrezerd,01r 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 10fgrzgrtz t Therefore for 01 and 02π we have d.ii10*fgegte t Let 01x for supiGgxe02π. Then G is the Hardy-Littlewood maximal function for g, and so lies in 0, 2πpL (, p. 12). Moreover, there is a constant Cp depending only on p such that pppLHGCg (In fact, for , ). Since 11pCp01 and 0, we obtain 1t  1001*supiixfg egxetd .G Hence 2π2π0011*d2π2πpippfg ed.ppppHGCg Therefore 12π0011sup* d2π.pppifgepHC 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 1:011ttz is the arc of the circle determined by 1, 11z, and 0, having endpoints 1 and 11z and not including the ori- gin. Since  is a probability measure, 101d1ttz 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  P. L. Duren, “Univalent Functions,” Springer-Verlag, New York, 1983.  D. J. Hallenbeck and T. H. MacGregor, “Linear Problems and Convexity Techniques in Geometric Function The- ory,” Pitman Publishing Ltd., London, 1984.  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.  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  P. L. Duren, “Theory of Hp Spaces,” Academic Press, New York, 1970.