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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.

Let B denote the set of all analytic functions

where the second dilatation function

is positive and hence, non-constant logharmonic mappings are sense-preserving and open on U. If f is a nonconstant logharmonic mapping of

where m is a nonnegative integer,

Note that

where

Let

for all

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

In Section 2, we determine the radius of starlikeness for the logharmonic mapping

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

Theorem 1 a) Let

b) If

Proof. a) Let

A simple calculations leads to

where

we obtain

This gives

Thus

b) Let

with

and also,

Hence,

and then simple calculations give that

Thus

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

Theorem 2 Let

i)

ii)

iii)

Equality holds for the right hand side if and only if

Proof. i) Let

For

and

Combining (2.2), (2.3) and (2.4), we get

Equality holds for the right hand side if and only if

For the left hand side inequality, we have

ii) and iii) Differentiation

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

Theorem 3 Let

Proof. Let

Now using (2.5) and (2.6) we have

Since

Substituting the bounds for

Theorem 4 of this section is an applications of the Baerstein star functions to the class of logharmonic mappings

where

If

One important property is that when

Other properties [

Lemma 1 For

b) For every

c) For every

Our main result of this section is the following theorem.

Theorem 4 If

Equality occurs if and only if

Proof. Let

where

Then

Write

As the star-function is sub-additive,

But since

each is subharmonic.

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

and

the later implies that

Proof. Let