Integral Means and Arc length of Starlike Log-harmonic Mappings

and Applied Analysis 3 One important property is that when u is a symmetric even rearrangement, then u∗ ( re ) ∫θ −θ u ( re ) dt. 2.6 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 g, h real and L1 on −a, a , the following are equivalent: a for every convex nondecreasing function Φ : R → R, ∫a −a Φ ( g x ) dx ≤ ∫a −a Φ h x dx; 2.7 b for every t ∈ R, ∫a −a ( g x − t dt ≤ ∫a −a h x − t dt; 2.8 c for every x ∈ 0, a , g∗ x ≤ g∗ x . 2.9 Here is the main result of the section. Theorem 2.2. If f z zh z g z H z g z is in STLh, then for each fixed r, 0 < r < 1, and as a function of θ a ( log|H z |∗ ≤ [ log [∣∣∣ z 1 − z ∣∣∣∣ exp ( Re 2z 1 − z )]]∗ , 2.10 b ( log ∣g z ∣)∗ ≤ [ log [ |1 − z| exp ( Re 2z 1 − z )]]∗ , 2.11 c ( log ∣f z ∣)∗ ≤ [ log [ |z| exp ( Re 4z 1 − z )]]∗ , 2.12 4 Abstract and Applied Analysis the three results are sharp by the functions f z z 1 − z 1 − z exp ( Re 4z 1 − z ) , zh z z 1 − z exp ( 2z 1 − z ) , g z 1 − z exp ( 2z 1 − z ) . 2.13 Proof. The proofs of the three parts are similar. We will emphasise the proof of part a . By 2.2 ,


Introduction
Let H U be the linear space of all analytic functions defined on the unit disk U {z : |z| < 1}.A Log-harmonic mapping is a solution to the nonlinear elliptic partial differential equation where the second dilatation function a ∈ H U such that |a z | < 1 for all z ∈ U.It has been shown that if f is a nonvanishing Log-harmonic mapping, then f can be expressed as where h and g are analytic functions in U. On the other hand, if f vanishes at z 0 but is not identically zero, then f admits the following representation: where Re β > −1/2, and h and g are analytic functions in U, g 0 1, and h 0 / 0 see 1 .Univalent Log-harmonic mappings have been studied extensively for details see 1-5 .
Let f z|z| 2β hg be a univalent Log-harmonic mapping.We say that f is starlike Logharmonic mapping if for all z ∈ U. Denote by ST * Lh the set of all starlike Log-harmonic mappings, and by S * the set of all starlike analytic mappings.It was shown in 4 that f z z|z| 2β h z g z ∈ ST * Lh if and only if ϕ z zh z /g z ∈ S * .In Section 2, using star functions we determine the integral means for starlike Logharmonic mappings.Moreover, we include the upper bound for the arc length of starlike Log-harmonic mappings.

Main Results
If f is univalent normalized starlike Log-harmonic mapping, then it was shown in 4 that where ϕ z H z /g z is starlike and a is analytic with a 0 0 and |a z | < 1 for z 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 f ∈ S, then where k z z/ 1 − z 2 , 0 < r < 1, and One important property is that when u 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 g, h real and L 1 on −a, a , the following are equivalent: 2.9 Here is the main result of the section.
Theorem 2.2.If f z zh z g z H z g z is in ST * Lh , then for each fixed r, 0 < r < 1, and as a function of θ

2.18
Consequently, by 2.4 and the fact that star functions respect subordination, dρ.

2.19
Abstract and Applied Analysis 5 Hence, as star functions are additive when functions are symmetric re-arrangements,

2.23
the later implies that f ∈ N hence has radial limits.
Proof.If we choose Φ x exp px 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 Φ x log x gives the second integral mean.
In the next theorem, we establish an upper for the arc length of starlike Log-harmonic mappings.

2.24
Now using 2.1 , we have

2.26
Since Re zϕ /ϕ is harmonic, and by the mean value theorem for harmonic functions, I 1 2π.Moreover, 1 a / 1 − a zϕ /ϕ is subordinate to 1 z / 1 − z 2 ; therefore, we have * ≤ log |1 − z| exp Re 2z 1 − z the three results are sharp by the functions