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
(1.1)
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
(1.2)
where m is a nonnegative integer, and, and are analytic functions in with and ([1] ). 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 [1] -[7] . 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 [8] -[14] . Such mappings are closely related to the theory of minimal surfaces, see [15] [16] .
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 [4] 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.
2. Basic Properties of Mappings from
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
(2.1)
A simple calculations leads to
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 [4]
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
(2.2)
For we have
(2.3)
and
(2.4)
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
(2.5)
and
(2.6)
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,
(2.7)
(2.8)
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 [17] [18] , [Chapter 7]. The first application was the remarkable result, if then
(3.1)
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
(3.2)
Other properties [18] , [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 [18] , [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
(3.3)
Write where is analytic, and (see [9] ).
As the star-function is sub-additive,
(3.4)
But since
each is subharmonic. is subordinate to and is subordinate to
Hence
and
Then,
Thus,
It follows that
(3.5)
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,
(3.6)
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.