A function f(z) defined on the unit disc U is said to be logharmonic if it is the solution of the nonlinear elliptic partial differential equation where such that . These mappings admit a global representation of the form where In this paper,we shall consider the logharmonic mappings , where is starlike. Distortion theorem and radius of starlikess are obtained. Moreover, we use star functions to determine the integral means for these mappings. An upper bound for the arclength is included.

A New Look for Starlike Logharmonic Mappings
1. Introduction

Let B denote the set of all analytic functions defined on the unit disk having the property that for all A logharmonic mapping defined on the unit disk is a solution of the nonlinear elliptic partial differential equation

where the second dilatation function. Because the Jacobian is positive and hence, non-constant logharmonic mappings are sense-preserving and open on U. If f is a nonconstant logharmonic mapping of and vanishes at but has no other zeros in U, then f admits the following representation

where m is a nonnegative integer, and, and are analytic functions in with and ( ). The exponent in (1.2) depends only on and can be expressed by Note that if and only if and that a univalent logharmonic mapping on vanish at the origin if and only if. Thus, a univalent logharmonic mappings on which vanishes at the origin will be of the form where and and have been studied extensively in the recent years, see  - . In this case, it follows that are univalent harmonic mappings of the half-plane

a detail study of univalent harmonic mappings to be found in  - . Such mappings are closely related to the theory of minimal surfaces, see   .

Let be a univalent logharmonic mapping. We say that is starlike logharmonic mapping if for all. Denote by the set of all starlike logharmonic mappings, and by the set of all starlike analytic mappings. It was shown in  that

if and only if

It is rather a natural question to ask whether there exists a linkage between the starlikeness of and

In Section 2, we determine the radius of starlikeness for the logharmonic mapping where A distortion theorem and an upper bound for the arclength of these mappings will be included.

In Section 3, we discuss the integral means for logharmonic mappings associated to starlike analytic mappings.

We start this section by establishing a linkage between the starlikeness of and

Theorem 1 a) Let be a logharmonic mapping where Then f maps the disk, where onto a starlike domain.

b) If. Then maps the disk, where onto a starlike domain.

Proof. a) Let be a logharmonic mapping with respect to and Suppose that Then can be written in the form where Since and

we obtain This gives Thus if Therefore, the radius of starlikeness is the smallest positive root (less than 1) of which is We conclude that f is univalent in and maps the disk onto a starlike domain.

b) Let be a starlike logharmonic mapping defined on the unit disk with respect to

with Then by 

and also, Hence, and then simple calculations give that Thus if Therefore, the radius of starlikeness is the smallest positive root (less than 1) of which is We conclude that is univalent in and maps the disk onto a starlike domain.

Our next result is a distortion theorem for the set of all logharmonic mappings where

Theorem 2 Let be a logharmonic mapping defined on the unit disk U where

then for

i)

ii)

iii)

Equality holds for the right hand side if and only if and which leads to

where Proof. i) Let be a logharmonic mapping with respect to with Suppose that Then can be written in the form

For we have

and

Combining (2.2), (2.3) and (2.4), we get Equality holds for the right hand side if and only if and which leads to For the left hand side inequality, we have ii) and iii) Differentiation in (2.2) with respect to and respectively leads to

and

The result follows from substituting from Theorem 2(i), (2.3) and (2.4) into (2.5) and (2.6).

In the next theorem we establish an upper bound for the arclength of the set of all logharmonic mappings where

Theorem 3 Let be a logharmonic mapping defined on the unit disk U where

Suppose that for then Proof. Let denote the closed curve which is the image of the circle under the mapping. Then Now using (2.5) and (2.6) we have Therefore,

Since is harmonic, and by the mean value theorem for harmonic functions, Also, is subordinate totherefore, we have Substituting the bounds for and in (2.8), we get 3. Integral Means

Theorem 4 of this section is an applications of the Baerstein star functions to the class of logharmonic mappings defined on the unit disk where. Star function was first introduced and properties were derived by Baerstein   , [Chapter 7]. The first application was the remarkable result, if then

where, and

If is a real function in an annulus then the definition of the star function of, is One important property is that when is symmetric (even) re-arrangement then

Other properties  , [Chapter 7] are that the star-function is sub-additive and star respects subordination. Respect means that the star of the subordinate function is less than or equal to the star of the function. In addition, it was also shown that star-function is additive when functions are symmetric re-arrangements. Here is a lemma, quoted in  , [Chapter 7] which we will use later.

Lemma 1 For real and on the following are equivalent a) For every convex non-decreasing function b) For every c) For every Our main result of this section is the following theorem.

Theorem 4 If be a logharmonic mapping defined on the unit disk U where then for each fixed and as a function of Equality occurs if and only if is one of the functions of the form, , where Proof. Let, then by (2.2), we have where and

Then

Write where is analytic, and (see  ).

But since each is subharmonic. is subordinate to and is subordinate to

Hence and Then, Thus, It follows that

Consequently, by combining (3.4), (3.5) and using the fact that star-functions respect subordination, it follows that Hence, as star-functions are additive when functions are symmetric re-arrangements, Now by using Theorem 4 we have Corollary 1 If be a logharmonic mapping defined on the unit disk U where then and the later implies that hence has radial limits.

Proof. Let this is non-decreasing convex function .The first integral mean can be obtained using part (a) of Lemma 1 and Theorem 4. Moreover, the choice yields the second integral mean.

ReferencesAbdulhadi Z. ,et al. (1996)Close-to-Starlike Logharmonic Mappings The International Journal of Mathematics and Mathematical Sciences 19, 563-574.Abdulhadi Z. ,et al. (2002)Typically Real Logharmonic Mappings The International Journal of Mathematics and Mathematical Sciences 31, 1-9.Abdulhadi Z. and Bshouty, D. ,et al. (1988)Univalent Functions in Transactions of the AMS—American Mathematical Society 305, 841-849.Abdulhadi, Z. and Hengartner, W. (1987) Spirallike Logharmonic Mappings. Complex Variables, Theory and Application, 9, 121-130.Abdulhadi Z., Hengartner W. and Szynal, J. ,et al. (1993)Univalent Logharmonic Ring Mappings Proceedings of the American Mathematical Society 119, 735-745.Abdulhadi Z. and Hengartner, W. ,et al. (1996)One Pointed Univalent Logharmonic Mappings Journal of Mathematical Analysis and Applications 203, 333-351.Abdulhadi, Z. and Hengartner, W. (2001) Polynomials in . Complex Variables, Theory and Application, 46, 89107.Abu-Muhanna, Y. and Lyzzaik, A. (1990) The Boundary Behaviour of Harmonic Univalent Maps. Pacific Journal of Mathematics, 141, 1-20. http://dx.doi.org/10.2140/pjm.1990.141.1Clunie J. and Sheil-Smal, T. ,et al. (1984)Harmonic Univalent Functions Annales Academic Scientiarum Fennice Mathematica 9, 3-25.Duren, P. and Schober, G. (1987) A Variational Method for Harmonic Mappings on Convex Regions. Complex Variables, Theory and Application, 9, 153-168. http://dx.doi.org/10.1080/17476938708814259Duren, P. and Schober, G. (1989) Linear Extremal Problems for Harmonic Mappings of the Disk. Proceedings of the American Mathematical Society, 106, 967-973. http://dx.doi.org/10.1090/S0002-9939-1989-0953740-5Hengartner, W. and Schober, G. (1986) On the Boundary Behavior of Orientation-Preserving Harmonic Mappings. Complex Variables, Theory and Application, 5, 197-208. http://dx.doi.org/10.1080/17476938608814140Hengartner, W. and Schober, G. (1986) Harmonic Mappings with Given Dilatation. Journal London Mathematical Society, 33, 473-483. http://dx.doi.org/10.1112/jlms/s2-33.3.473Jun, S.H. (1993) Univalent Harmonic Mappings on Proceedings of the American Mathematical Society, 119, 109-114. http://dx.doi.org/10.1090/S0002-9939-1993-1148026-3Nitsche, J.C.C. (1989) Lectures on Minimal Surfaces. Vol. 1, Translated from the German by Jerry M. Feinberg, Cambridge University Press, Cambridge.Osserman, R. (1986) A Survey of Minimal Surfaces. 2nd Edition, Dover, New York.Baernstein, A. (1974) Integral Means, Univalent Functions and Circular Symmetrization. Acta Mathematica, 133, 139169. http://dx.doi.org/10.1007/BF02392144Duren, P. (1983) Univalent Functions. Springer-Verlag, Berlin.