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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 721943, 32 pages
Univalent Logharmonic Mappings in the Plane
1Department of Mathematics, American University of Sharjah, P.O. Box 26666, Sharjah, UAE
2School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia
Received 29 September 2011; Accepted 8 December 2011
Academic Editor: Saminathan Ponnusamy
Copyright © 2012 Zayid Abdulhadi and Rosihan M. Ali. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper surveys recent advances on univalent logharmonic mappings defined on a simply or multiply connected domain. Topics discussed include mapping theorems, logharmonic automorphisms, univalent logharmonic extensions onto the unit disc or the annulus, univalent logharmonic exterior mappings, and univalent logharmonic ring mappings. Logharmonic polynomials are also discussed, along with several important subclasses of logharmonic mappings.
Let be a domain in the complex plane . Denote by (resp., by ) the linear space of all analytic (resp., meromorphic) functions in , and let be the set of all functions satisfying , . A nonconstant function is logharmonic in if is the solution of the nonlinear elliptic differential equation. The function is called the second dilatation of . In contrast to the linear space consisting of analytic functions, translations in the image do not preserve logharmonicity, and the inverse of a logharmonic function is not necessarily logharmonic. If and are two logharmonic functions with respect to , then is logharmonic with respect to the same . If, in addition, , then is also logharmonic. The composition of a logharmonic mapping with a conformal premapping is also logharmonic with respect to . However, the composition of a conformal postmapping with a logharmonic mapping is in general not logharmonic. If is a logharmonic mapping in , then is a nonconstant locally quasiregular mapping, and, therefore, it is continuous, open, and light. It follows that can be represented as a composition of two functions , where is a locally quasiconformal homeomorphism in and . As an immediate consequence, the maximum principle, the identity principle, and the argument principle all still hold for logharmonic mappings.
A local representation for logharmonic mappings was given by Abdulhadi and Bshouty in . In particular, they obtained the following result.
Theorem 1.1. Let be a logharmonic mapping in with respect to . Suppose that and , where and . Then admits the representation where , and, therefore, . The functions and are in , with and .
As a direct consequence of Theorem 1.1, we have the following global representation for logharmonic mappings.
Corollary 1.2. Let be a simply connected domain in and a logharmonic mapping in . If has exactly zeros in (counting multiplicities), then admits a global representation given by where and, therefore, . The functions and are in , and .
For the converse, Abdulhadi and Hengartner  proved the following theorem.
Theorem 1.3. Suppose that is defined in a domain , where and are in , such that does not lie on a logarithmic spiral. Then either or is a solution of
Remark 1.4. The converse of Theorem 1.3 does not hold. Indeed, consider the partial differential equation . Then and are solutions of this equation. The function can be written in the form while could not.
Remark 1.5. The function , cannot be written in the form unless or is a constant. However, it is a solution of the second Beltrami equation where Hence, in and is independent of .
Corollary 1.6. The image of a nonconstant function lies on a logarithmic spiral if and only if is a solution of (1.1) with .
In the theory of quasiconformal mappings, it is proved that, for any measurable function with , the solution of Beltrami equation can be factorized in the form , where is a univalent quasiconformal mapping and is an analytic function (see ). For sense-preserving harmonic mappings, the answer is negative. In , Duren and Hengartner gave a necessary and sufficient condition on sense-preserving harmonic mappings for the existence of such a factorization. Moreover, for logharmonic mappings, such a factorization need not exist. For example, the function is a sense-preserving logharmonic mapping with respect to , and has no decomposition of the desired form (see ). The following factorization theorem was proved in .
Theorem 1.7. Let be a nonconstant logharmonic mapping defined in a domain , and let be its second dilatation function. Then can be factorized in the form , for some analytic function and some univalent logharmonic mapping if and only if(a) in ,(b) implies .
Under these conditions, the representation is unique up to a conformal mapping; any other representation has the form and for some conformal mapping defined in .
Consider now the logharmonic mapping . The point is an isolated singularity of , and is continuous at this point. However, does not admit a logharmonic-continuation to . A further restriction is needed.
Theorem 1.8 (see  (logharmonic-continuation across an isolated singularity)). Let be the point disc , and let defined in be a logharmonic mapping with respect to satisfying . Then admits a logharmonic-continuation across the origin and has the representation where and are nonnegative integers, , and and are analytic functions on satisfying .
Liouville’s theorem does not hold for entire logharmonic functions. The function is a nonconstant bounded logharmonic in . Its dilatation is . However, the following modified version of Liouville’s theorem was given in .
Theorem 1.9 (modified Liouville’s theorem). Let be a bounded logharmonic function in . Then either the image is a circle centered at the origin with dilatation function or is a constant.
Let be a logharmonic mapping defined in a domain with respect to satisfying . Let(1),(2),(3),(4) be the cardinality of , that is, the number of zeros of in , multiplicity is not counted,(5) be the number of zeros of in , multiplicity counted.
The following argument principle for logharmonic mappings in is shown in .
Theorem 1.10 (generalized argument principle for logharmonic mappings). Let be a Jordan domain, and let be a logharmonic mapping defined in the closure with respect to satisfying . Fix such that is empty. Then
As a consequence of the argument principle, the following result is obtained.
Theorem 1.11. Let be a sequence of logharmonic mappings defined in with respect to a given , where is the unit disc. Suppose that converges locally uniformly to and that converges locally uniformly to a logharmonic mapping with respect to . If for all , then .
In Section 2, a survey is given on univalent logharmonic mappings defined in a simply connected domain of . Section 3 deals with univalent logharmonic mappings defined on multiply connected domains, while Section 4 considers logharmonic polynomials. The final section of the survey discusses several important subclasses of logharmonic mappings.
2. Univalent Logharmonic Mappings in a Simply Connected Domain
Let be a domain in the complex plane , and let be a nonparametric minimal surface lying over . Then can be represented by a function , , and there is a univalent orientation-preserving harmonic mapping from an appropriate domain of onto which determines in the following sense. The mapping is a solution of the system of linear elliptic partial differential equation where . Since is orientation preserving, it follows that in . The function is the second dilatation of . The value is the quotient of the maximum value and the minimum value of the differential when varies on the unit circle (see, e.g., [12, 13]). The representation of the minimal surface is given by three real-valued harmonic functions (see, e.g., [13, 14]),
Since in , it follows that belongs to . In particular, each zero of is of even order. Since the Riemannian metric of is , it follows that and are isothermal parameters for . Moreover, the exterior unit normal vector , to the minimal surface (known as the Gauss mapping) depends only on the second dilatation function of . More precisely, The inverse of the stereographic projection of the Gauss mapping called the Weierstrass parameter.
The following question arises: What are the domains ? If is univalent and analytic and if is univalent and harmonic, then the composition (whenever well defined) is a univalent harmonic mapping but need not be harmonic. Hence, if represents a minimal surface over (in the sense of relation (2.2)), then represents the same minimal surface but in other isothermal parameters.
Suppose that is a proper simply connected domain in . Then, we may choose for any proper simply connected domain in . In particular, or are appropriate choices.
Consider now the left half-plane , and let be a univalent harmonic and orientation-preserving map defined in satisfying the relation where and are real constants. Applying the transformation , it may be assumed without loss of generality that , that is, Whenever for some , we will write . Similarly, means that .
Let UHP denote the class of all univalent harmonic orientation-preserving mappings defined on the left half-plane satisfying It follows that the second dilatation function is periodic, that is, in , and therefore the Gauss map is also periodic. Observe that exists. Furthermore, it was shown in  that mappings in the class UHP admit the representation where(a) and are in such that(i) and exists and finite in ,(ii) and for all ;(b)
Define Then is a univalent logharmonic mapping in with respect to and hence . Observe that the family of all univalent logharmonic and orientation-preserving mappings defined in satisfying is isomorphic to the class . It was shown in [4, 7] that it is easier to work with logharmonic mappings even if the differential equation becomes nonlinear.
2.2. Univalent Logharmonic Mappings
Let be a simply connected domain in , , and suppose that is a univalent logharmonic mapping defined in . If , then is a univalent and harmonic mapping in . This mapping has been extensively studied in [15–18]. If and is a univalent logharmonic mapping defined in , then the representation (1.2) of becomes for every , where(a), and so ,(b) and are in satisfying and .
It follows that is locally quasiconformal. The analogue of Caratheodory’s Kernel Theorem might fail for univalent logharmonic mappings. Indeed, each function which is univalent and logharmonic with respect to , satisfies the normalization , , and maps the unit disc onto the slit domain . The tip of the omitted slit varies monotonically from to −1 as varies from 0 to 1. The limit function is univalent and logharmonic and maps onto . It has the boundary value for , and the cluster set of at the point 1 is the unit circle.
Let be a simply connected domain in and . The following characterization theorem was proved in .
Theorem 2.1. Let be a univalent mapping defined in such that . Then is of the form if and only if is a logharmonic mapping with respect to satisfying , .
Univalent logharmonic mappings have the following properties.
Theorem 2.2 (see ). Let be a simply connected domain in and a univalent logharmonic mapping defined in with respect to .(a)Then for all whenever .(b)If , then exists and is in . Therefore, is a nonvanishing function in .(c)Let be a complex number such that . Then is a univalent logharmonic mapping with respect to
There are few logharmonic mappings that are univalent on the whole complex plane . Indeed, Abdulhadi and Bshouty  showed the following.
Theorem 2.3. A function is a univalent logharmonic mapping defined in with respect to if and only if
Now let be a simply connected proper domain in and a univalent logharmonic function in with respect to . Denote by a conformal mapping from the unit disc onto . Then is univalent logharmonic in with respect to . Therefore, we may assume that and .
Analogous to the analytic case, we denote Now , and is not compact with respect to the topology of normal convergence. Indeed, the sequence is in , and it converges uniformly to not in . Our next result deals with the subclass of defined by (resp., . The following result was proved in .
Theorem 2.4. is compact in the topology of normal locally uniform convergence.
Remark 2.5. In contrast to univalent harmonic mappings, is not a normal family. Indeed, is not locally uniformly bounded for sufficiently large.
The following interesting distortion theorem is due to Abdulhadi and Bshouty , and it was used in the proof of the mapping theorem.
Theorem 2.6. If , then . In particular, the disc is in .
2.3. Mapping Theorem
We look for an analogue of the Riemann Mapping Theorem. Let be a simply connected domain in , and let be given. Fix and . We are interested in the existence of a univalent logharmonic function from into with respect to the given function and normalized by and . If for all , then the univalent logharmonic mappings are quasiconformal, and therefore the problem is solvable.
Suppose that we want to find a univalent logharmonic mapping with , normalized by and such that maps onto . Assume that such a function exists. Then, using Theorem 5.1 , it follows that must be of the form Observe that is univalent in , but maps onto a disc, and not onto a slit domain. In other words, there is no univalent logharmonic mapping defined in with respect to satisfying , and . However, the following mapping theorem was proved in .
Theorem 2.7. Let be a bounded simply connected domain in containing the origin, and whose boundary is locally connected. Let be given. Then there is a univalent logharmonic function defined in with the following properties.(i) is a solution of (1.1).(ii), normalized at the origin by , where and .(iii) exists and is in for each , where is a countable set.(iv)For each and exist and are in .(v)For , the cluster set of at lies on a helix joining the point to the point .
Remark 2.8. In the case where , properties (ii) and (iii) imply that .
Remark 2.9. If and , then the cluster set at is a circle. Suppose that , then there are infinitely many helices joining and . But the cluster set of at lies on one of them. For example, the cluster set of at lies on the helix, joining the points and , where the cluster set of at is the straight line segment from and .
The uniqueness of the mapping theorem was proved in  for the special case is a strictly starlike and bounded domain; that is, every ray starting at the origin intersects at exactly one point.
Theorem 2.10 (uniqueness in the mapping theorem). Let be given such that . Let be a strictly starlike and bounded domain. Then there exists a unique univalent logharmonic function with respect to such that and .
2.4. Logharmonic Automorphisms
We consider univalent logharmonic mappings from onto . With no loss of generality, it is assumed that and . Otherwise, we consider an appropriate Möbius transformation of the preimage. Let denote the class of such mappings. The following two theorems established in  characterize completely mappings in .
Theorem 2.11. Let and be two nonvanishing analytic functions in . Then is in satisfying and if and only if , and in .
We now associate to each in with the mapping .
(a) For each and for each , , there is one and only one such that for every and .
(b) For each , there is a unique solution of (1.1) which is in .
Remark 2.13. Part (a) of Theorem 2.12 is quite surprising. Indeed, consider and . Then , almost everywhere; however, . To be more precise, the corresponding mapping is satisfying for all , where the cluster set of at the point 1 is the unit circle.
2.5. Univalent Logharmonic Mappings Extensions onto the Unit Disc
In 1926 Kneser  obtained the following result.
Theorem 2.14. Let be a bounded simply connected Jordan domain, and let be an orientation-preserving homeomorphism from the unit disc circle onto . Then, if , the solution of the Dirichlet problem the Poisson integral is univalent on the unit disc .
Theorem 2.15. Let be a homeomorphism from onto , where is a bounded convex domain. Then the Dirichlet solution is univalent on .
We will use the following definition.
Definition 2.16. Let be the unit disc or the annulus , , and suppose that is a continuous function defined on . One says that is a logharmonic solution of the Dirichlet problem if(a) is a solution of the form (1.1),(b) is continuous in ,(c).
Theorem 2.17. Let be a nonvanishing continuous complex-valued function defined on . Then there exist and analytic in which are independent of , such that is a logharmonic solution of the Dirichlet problem . Furthermore, if , then and are uniquely determined.
Theorem 2.18. Let be an orientation-preserving homeomorphism from onto , where is continuous and strictly monotonically increasing on . Furthermore, suppose that . Then, for a given with , the logharmonic solution of the Dirichlet problem which is of the form is univalent in .
2.6. Boundary Behavior
Let be a univalent logharmonic mapping in the unit disc with respect to . If for all , then is a quasiconformal map, and its boundary behavior is the same as for conformal mappings. However, if approaches one as tends to the boundary, then the boundary behavior of is quite different. It may happen that the boundary values are constant on an interval of , or that there are jumps as the following example shows.
Example 2.19. The function is a univalent logharmonic mapping in the unit disc with respect to , such that . It follows that for all and that the cluster set of at the point 1 is the unit circle.
The following theorem was stated in .
Theorem 2.20. Let be a simply connected domain of whose boundary is locally connected, and . Let be a univalent logharmonic mapping from onto satisfying . Then the nonrestricted limit of at exists on , where is a countable set. If , then jumps at , and the cluster set at is a subinterval of a logarithmic spiral.
The next theorem  shows that the boundary values of depend strongly on the values of .
Theorem 2.21. Let be a simply connected domain of whose boundary is locally connected and . Suppose that the function has an analytic extension across an open subinterval of the unit circle , such that in . Let be a univalent logharmonic mapping with respect to which maps onto and satisfies . Then the following relations hold in .(a)Let and be a continuous function on the set . If , then (b)If is continuous at , then (c)If jumps at , which must and can happen only when lies on a segment of a logarithmic spiral, for , then (d)If is not constant on a subinterval of , then the right limit exists everywhere on .
2.7. A Constructive Method
In this section, a method is introduced for constructing univalent logharmonic mappings from the unit disc onto a strictly starlike domain , which has been successfully applied to conformal mappings (see, e.g., [23–25]), as well as for univalent harmonic mappings (see, e.g., [26, 27]).
Let be a strictly starlike domain of . Then can be expressed in the parametric form where is a positive continuous function on . The following notations will be used:
For all , define Then
For any complex-valued function in , define The following properties are due to Bshouty et al. .
The next lemma shows that is lower semicontinuous with respect to the point-wise convergence; this was proved in .
Lemma 2.23. Let be a strictly starlike domain of , and let be a sequence of mappings from into which converges pointwise to . Then . Strict inequality can hold even in the case of locally uniform convergence.
Let be a fixed strictly starlike domain of , and let be a given (second) dilatation function. Denote by the set of all logharmonic mappings with respect to the given dilatation function which are normalized by . Observe that since it is assumed that . Hengartner and Nadeau  solved the following optimization problem.
Theorem 2.24. Let be a strictly starlike domain of , and let , be given. Denote by the univalent logharmonic mapping satisfying , and . Then there exists a unique such that for all and .
Theorem 2.24 allows us to solve the following mathematical program: For , , where and Furthermore, each is an open mapping. Denote by the set of all mappings of the form and by any solution of the optimization problem
Theorem 2.25. Let be a polynomial such that in . Then the sequence of solutions of converges locally uniformly to the univalent solution of
The question remains how big could be. It follows from Theorem 5.24 that and . Suppose that is a Jordan domain whose boundary is rectifiable and piecewise smooth. Hengartner and Nadeau  obtained the following additional estimate for the coefficients.
Theorem 2.26. Let be a univalent logharmonic mapping from onto , and let be the length of . Then Equality holds for the case and .
3. Univalent Logharmonic Mappings on Multiply Connected Domains
3.1. Univalent Logharmonic Exterior Mappings
This section considers univalent logharmonic and orientation-preserving mappings defined on the exterior of the unit disc , , satisfying . These mappings are called univalent logharmonic exterior mappings. If does not vanish on , then is a univalent logharmonic mapping defined in normalized by . Moreover, is a univalent harmonic mapping defined on the right half-plane satisfying the relation and is a solution of the linear elliptic partial differential equation where the second dilatation function , satisfies on . Such mappings were studied in [9, 29–32]. Several authors have also studied harmonic mappings between Riemannian manifolds, and an excellent survey has been given in [33–37].
The next result proved in  is a global representation of univalent logharmonic exterior mappings.
Theorem 3.1. Let be a univalent logharmonic mapping defined on the exterior of the closed unit disc such that Suppose that for some , or if does not vanish, let . Then there are two complex numbers and , , and two nonvanishing analytic functions and on such that , and is of the form for all .
Let be a univalent logharmonic exterior mapping defined on the exterior of the closed unit disc such that . Applying an appropriate rotation to the preimage, we may assume that .
Definition 3.3. The class consists of all univalent logharmonic mappings defined on which are of the form (3.2), where , and and are analytic nonvanishing functions on , normalized by and .
Let be a univalent logharmonic mapping in with . Then there is a real number and a positive constant such that belongs to . If does not vanish on , then the set of omitted values is a continuum. In other words, there is no univalent logharmonic mapping defined on satisfying and . Note that 0 is an exceptional point, since, for each , there are univalent logharmonic mappings such that . Assume that , let , and let be an omitted value of . Applying a rotation to the image , we may assume that , and we restrict ourselves to the subclass .
In the next theorem, Abdulhadi and Hengartner  gave a complete characterization of all mappings in the class .
Theorem 3.4. A mapping belongs to and if and only if is of the form where and satisfy the inequality
3.2. Univalent Logharmonic Ring Mappings
In this section we investigate the family of all univalent logharmonic mappings which map an annulus , onto an annulus for some satisfying the condition for all . The last condition says that the outer boundary corresponds to the outer boundary. We call an element a univalent logharmonic ring mapping.
If , then and , are the only mappings in . In the case of univalent harmonic mappings from onto , it may be possible that ; for example, has this property. However, Nitsche  has shown that there is an such that there is no univalent harmonic mapping from onto whenever .
There is no univalent logharmonic mappings from , , onto . This is a direct consequence of Theorem 3.5. But, on the other hand, for there is neither a positive lower bound nor a uniform upper bound strictly less than one. Indeed, , is univalent on , and its image is .
Unlike the case of univalent harmonic mappings, univalent logharmonic mappings need not have a continuous extension onto the closure of . Indeed, is a univalent logharmonic ring mapping from onto itself whose cluster sets on the outer boundary are , if , and .
Let be the set of all univalent analytic functions on with the properties(i),(ii) on .
Theorem 3.5. A function belongs to if and only if where(a) and ,(b) on ,(c), , (d).
In particular, functions belonging to map concentric circles onto concentric circles.
Theorem 3.6. A function is in if and only if it is of the form where and .
Next we fix the second dilatation function for all . The following existence and uniqueness theorem was established in .
Theorem 3.7. For a given for all , and, for a given , there exists one and only one univalent solution of (1.1) in such that .
3.3. Univalent Logharmonic Mappings Extensions onto the Annulus
The next two theorems proved in  deal with the solution of the Dirichlet problem for ring domains.
Theorem 3.9. Let be a nonvanishing continuous function defined on the boundary of the annulus . Then there exists, for each , a unique mapping of the form (2.10), which is continuous on the closure of and satisfies on .
Theorem 3.10. Let and , be a given continuous function on , satisfying(a) and on ,(b).
Then the logharmonic solution of the Dirichlet problem with respect to and is a univalent mapping from onto .
4. Logharmonic Polynomials
Denote by an analytic polynomial of degree . A logharmonic polynomial is a function of the form . In contrast to the analytic case, there are nonconstant logharmonic polynomials which are not -valent for every . For example, the function is a logharmonic polynomial in with respect to . Moreover, the function is a logharmonic polynomial in with respect to . This polynomial is two-valent and omits the half-plane . On the other hand, they inherit the property of analytic polynomials. This follows from the fact that . However, the converse is not true; there are logharmonic functions defined in which are not logharmonic polynomials and have the property . The function is such an example. Note that there are harmonic polynomials which do not satisfy . However, if it is assumed that exists and , then the following result  is deduced.
Theorem 4.1. Let be a logharmonic function in such that . If exists and if , then is a polynomial.
Denote by the cardinality of , that is, the number of zeros of in , multiplicity not counted. The polynomial has the property that . On the other hand, using Theorem 2.3, it follows that a univalent logharmonic mapping in is necessarily a polynomial which is either of the form or of its conjugate, where , and . There are functions of the form which are not polynomials but have the property that is finite and uniformly bounded for all . For example, the function has at most two zeros for all fixed . The following result was shown in .
Theorem 4.2. Let be a logharmonic function in such that is finite for at least two different values of , exists with , then is a polynomial.
An upper bound on the number of -points of a logharmonic polynomial can be readily obtained by using Bezout’s theorem .
Theorem 4.3 (see ). Let and be polynomials in the real variables and with real coefficients. If and , then either and have at most common zeros or they have infinitely many zeros.
Wilmshurst  has shown that Bezout’s theorem gives a sharp upper bound for the number of zeros of a harmonic polynomial and hence for polyanalytic polynomials (see, e.g., [41, 42]). However, this is not true for logharmonic polynomials.
Let be a logharmonic polynomial of degree . Then . The functions and are real-valued polynomials in and and are of degree . Applying Bezout’s theorem, we conclude with the following estimate.
Theorem 4.4. Let be a logharmonic polynomial defined in . Then either has infinitely many zeros or has at most zeros for all .
The bound is not the best possible. Indeed, a quadratic polynomial is of the form , , or . In all three cases, has either infinitely many zeros or it has at most two.
Observe that the logharmonic polynomial is 2-valent and omits the half-plane and that . However, the situation changes if and we have the following result .
Theorem 4.5. Let be a logharmonic polynomial defined in , and suppose that . Fix such that is empty. Then the number of zeros of , and hence also the valency of in , is at least . The bound is best possible.
The following result is an immediate consequence of Theorem 4.5.
Corollary 4.6. Let be a logharmonic polynomial defined in , and suppose that . Then(i), (ii)for almost all , the function has at least disjoint zeros.
The next result characterizes polynomials of finite valency .
Theorem 4.7. Let be a logharmonic polynomial defined in , such that . Then the cardinality of the zero set is finite (hence, by Bezout’s theorem, uniformly bounded) for all .
Remark 4.8. If , then the image lies on a straight line.
5. Subclasses of Logharmonic Mappings
5.1. Spirallike Logharmonic Mappings
Let be a simply connected domain if contains the origin. We say that is -spirallike, , if implies that for all . If , the domain is called starlike (with respect to the origin). We will use the following notations.(a) is the set of all univalent logharmonic mappings in satisfying , , and is an -spirallike domain.(b) and .(c) and , for which is starlike (with respect to the origin).
To each , we associate the analytic function , . Abdulhadi and Hengartner  gave a representation theorem for mappings in the class .
(a) If , then .
(b)For any given and , there are and in uniquely determined such that(i), , (ii), (iii) is a solution of (1.1) in , where .
Remark 5.2. Theorem 5.1 has no equivalence for the class of all convex univalent logharmonic mappings. Indeed, is a convex mapping, , but is not a convex mapping.
Remark 5.3. Theorem 5.1 asserts that the class fixed in , is isomorphic to .
The following result is an immediate consequence of Theorem 5.1.
Corollary 5.4. If , then for all . In other words, level sets inherit the property of being -spirallike.
The next result is an integral representation for .
Theorem 5.5. A function if and only if there are two probability measures and on the Borel sets of and an such that