Abstract and Applied Analysis

Volume 2011, Article ID 193184, 7 pages

http://dx.doi.org/10.1155/2011/193184

## Integral Means and Arc length of Starlike Log-harmonic Mappings

Department of Mathematics, American University of Sharjah, P.O. Box 26666 Sharjah, United Arab Emirates

Received 13 January 2011; Accepted 17 February 2011

Academic Editor: Gaston Mandata N'Guerekata

Copyright © 2011 Z. Abdulhadi and Y. Abu Muhanna. 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.

#### Abstract

We use star functions to determine the integral means for starlike log-harmonic mappings. Moreover, we include the upper bound for the arc length of starlike log-harmonic mappings.

#### 1. Introduction

Let be the linear space of all analytic functions defined on the unit disk . A Log-harmonic mapping is a solution to the nonlinear elliptic partial differential equation where the second dilatation function such that for all . It has been shown that if is a nonvanishing Log-harmonic mapping, then can be expressed as where and are analytic functions in . On the other hand, if vanishes at but is not identically zero, then admits the following representation: where , and and are analytic functions in , , and (see [1]). Univalent Log-harmonic mappings have been studied extensively (for details see [1–5]).

Let be a univalent Log-harmonic mapping. We say that is starlike Log-harmonic mapping if for all . Denote by the set of all starlike Log-harmonic mappings, and by the set of all starlike analytic mappings. It was shown in [4] that if and only if .

In Section 2, using star functions we determine the integral means for starlike Log-harmonic mappings. Moreover, we include the upper bound for the arc length of starlike Log-harmonic mappings.

#### 2. Main Results

If is univalent normalized starlike Log-harmonic mapping, then it was shown in [4] that where is starlike and is analytic with and for in the unit disk,

Theorem 2.2 of this section is an application of the Baerstein star functions to starlike Log-harmonic mapping. Star function was first introduced and properties were derived by Baerstein [6], [7, 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 a symmetric (even) rearrangement, then

Other properties [6], [7, Chapter 7] are that the star function is subadditive and the 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 rearrangements. Here is a lemma, quoted in [6], [7, Chapter 7], which we will be used later.

Lemma 2.1. *For real and on , the following are equivalent: *(a)*for every convex nondecreasing function ,
*(b)*for every ,
*(c)*for every ,
**Here is the main result of the section.*

Theorem 2.2. *If is in , then for each fixed , and as a function of *(a)*(b)**(c)**
the three results are sharp by the functions
*

*Proof. *The proofs of the three parts are similar. We will emphasise the proof of part (a).

By (2.2),

Then
where . Write , where is analytic, , and , (see [7]). Then, as is subordinate to [7],
for analytic, and .

Then (2.15) becomes

As the star function is subadditive,

Consequently, by (2.4) and the fact that star functions respect subordination,
Hence, as star functions are additive when functions are symmetric re-arrangements,

Therefore,
which is part (a).

By (2.4), (2.15) and in similar fashion to the upper part,
(2.21) and (2.22) give part (c).

Now by using Lemma 2.1, we have the following corollary.

Corollary 2.3. *If is starlike Log-harmonic, then, for ,
**
the later implies that hence has radial limits.*

*Proof. *If we choose which is nondecreasing convex, then part (a) of Lemma 2.1 and part (c) of Theorem 2.2 give the first integral mean. The choice gives the second integral mean.

In the next theorem, we establish an upper for the arc length of starlike Log-harmonic mappings.

Theorem 2.4. *If and for , then .*

*Proof. *Let denote the closed curve which is the image of the circle under the mapping . Then
Now using (2.1), we have

Therefore,

Since is harmonic, and by the mean value theorem for harmonic functions, . Moreover, is subordinate to ; therefore, we have

Substituting the bounds for and in (2.26), we get

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