ABSTRACT

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.

Keywords:Logharmonic, Univalent, Starlike, Integral Means, Arclength

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.

References