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Abstract and Applied Analysis
Volume 2011, Article ID 193184, 7 pages
http://dx.doi.org/10.1155/2011/193184
Research Article

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 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 𝑓𝑧𝑓𝑓=𝑎𝑧𝑓,(1.1) where the second dilatation function 𝑎𝐻(𝑈) such that |𝑎(𝑧)|<1 for all 𝑧𝑈. It has been shown that if 𝑓 is a nonvanishing Log-harmonic mapping, then 𝑓 can be expressed as 𝑓(𝑧)=(𝑧)𝑔(𝑧),(1.2) where 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: 𝑓(𝑧)=𝑧|𝑧|2𝛽(𝑧)𝑔(𝑧),(1.3) where Re𝛽>1/2, and and 𝑔 are analytic functions in 𝑈, 𝑔(0)=1, and (0)0 (see [1]). Univalent Log-harmonic mappings have been studied extensively (for details see [15]).

Let 𝑓=𝑧|𝑧|2𝛽𝑔 be a univalent Log-harmonic mapping. We say that 𝑓 is starlike Log-harmonic mapping if 𝜕arg𝑓𝑟𝑒𝑖𝜃𝜕𝜃=Re𝑧𝑓𝑧𝑧𝑓𝑧𝑓>0,(1.4) for all 𝑧𝑈. Denote by STLh 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(𝑧)𝑔(𝑧)STLh if and only if 𝜑(𝑧)=𝑧(𝑧)/𝑔(𝑧)𝑆.

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 𝑓 is univalent normalized starlike Log-harmonic mapping, then it was shown in [4] that 𝑓(𝑧)=𝑧(𝑧)𝑔(𝑧)=𝐻(𝑧)𝑔(𝑧)=𝜑(𝑧)exp2Re𝑧0𝑎(𝑠)𝜑1𝑎(𝑠)(𝑠)𝜑(𝑠)𝑑𝑠,(2.1) 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, 𝐻(𝑧)=𝜑(𝑧)exp𝑧0𝑎(𝑠)𝜑1𝑎(𝑠)(𝑠)𝜑(𝑠)𝑑𝑠,(2.2)𝑔(𝑧)=exp𝑧0𝑎(𝑠)𝜑1𝑎(𝑠)(𝑠)𝜑(𝑠)𝑑𝑠.(2.3)

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 𝑓𝑆, then ||𝑓𝑟𝑒𝑖𝑡||𝑝||𝑘𝑑𝑡𝑟𝑒𝑖𝑡||𝑝𝑑𝑡,(2.4) where 𝑘(𝑧)=𝑧/(1𝑧)2, 0<𝑟<1, and 𝑝>0.

If 𝑢(𝑧) is a real 𝐿1 function in an annulus 0<𝑅1<|𝑧|<𝑅2, then the definition of the star function of 𝑢, 𝑢 is 𝑢𝑟𝑒𝑖𝜃=sup||𝐸||=2𝜃𝐸𝑢𝑟𝑒𝑖𝑡𝑑𝑡,for𝑅1<𝑟<𝑅2.(2.5)

One important property is that when 𝑢 is a symmetric (even) rearrangement, then 𝑢𝑟𝑒𝑖𝜃=𝜃𝜃𝑢𝑟𝑒𝑖𝑡𝑑𝑡.(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, real and 𝐿1 on [𝑎,𝑎], the following are equivalent: (a)for every convex nondecreasing function Φ𝑅𝑅, 𝑎𝑎Φ(𝑔(𝑥))𝑑𝑥𝑎𝑎Φ((𝑥))𝑑𝑥;(2.7)(b)for every 𝑡𝑅, 𝑎𝑎(𝑔(𝑥)𝑡)+𝑑𝑡𝑎𝑎((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)=𝑧(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)||||log𝐻(𝑧)||||𝑧log||||(1𝑧)expRe2𝑧1𝑧,(2.10)(b)||||log𝑔(𝑧)||log1𝑧||expRe2𝑧1𝑧,(2.11)(c)||||log𝑓(𝑧)log|𝑧|expRe4𝑧1𝑧,(2.12)
the three results are sharp by the functions 𝑧𝑓(𝑧)=1𝑧(1𝑧)expRe4𝑧,𝑧1𝑧𝑧(𝑧)=(1𝑧)exp2𝑧,1𝑧𝑔(𝑧)=(1𝑧)exp2𝑧.1𝑧(2.13)

Proof. The proofs of the three parts are similar. We will emphasise the proof of part (a).
By (2.2), 𝐻(𝑧)=Φ(𝑧)exp10𝑎(𝜌𝑧)1𝑎(𝜌𝑧)𝑧Φ(𝜌𝑧)Φ(𝜌𝑧)𝑑𝜌.(2.14)
Then ||||||||log𝐻(𝑧)=logΦ(𝑧)+Re10𝑎(𝜌𝑧)1𝑎(𝜌𝑧)𝑧Φ(𝜌𝑧)Φ(𝜌𝑧)𝑑𝜌,(2.15) where 𝑧=𝑟𝑒𝑖𝜃. Write (𝑎(𝜌𝑧)/(1𝑎(𝜌𝑧)))((𝜌𝑧)Φ(𝜌𝑧)/Φ(𝜌𝑧))=(𝑎(𝜌𝑧)/(1𝑎(𝜌𝑧)))((1+𝜔(𝜌𝑧))/𝜌(1𝜔(𝜌𝑧))), where 𝜔 is analytic, |𝜔|<1, and 𝜔(0)=0, (see [7]). Then, as (𝑎(𝑧)/(1𝑎(𝑧)))(𝑧Φ(𝑧)/Φ(𝑧)) is subordinate to (𝑧(1+𝑧)/(1𝑧)2) [7], 𝑎(𝑧)1𝑎(𝑧)(𝑧)Φ(𝑧)=Φ(𝑧)𝜓(𝑧)(1+𝜓(𝑧))(1𝜓(𝑧))2,(2.16) for 𝜓 analytic, |𝜓|<1 and 𝜓(0)=0.
Then (2.15) becomes ||||||||log𝐻(𝑧)=logΦ(𝑧)+Re101(𝜌)𝜓(𝜌𝑧)(1+𝜓(𝜌𝑧))(1𝜓(𝜌𝑧))2𝑑𝜌.(2.17)
As the star function is subadditive, ||||log𝐻(𝑧)||||logΦ(𝑧)+101Re𝜌𝜓(𝜌𝑧)1+𝜓(𝜌𝑧)(1𝜓(𝜌𝑧))2𝑑𝜌.(2.18)
Consequently, by (2.4) and the fact that star functions respect subordination, (log|𝐻(𝑧)|)||||𝑧log(1𝑧)2||||+101𝜌Re𝜌𝑧(1+𝜌𝑧)(1𝜌𝑧)2𝑑𝜌.(2.19) Hence, as star functions are additive when functions are symmetric re-arrangements, (log|𝐻(𝑧)|)||||𝑧log(1𝑧)2||||+101𝜌Re𝜌𝑧(1+𝜌𝑧)(1𝜌𝑧)2=||||𝑧𝑑𝜌log(1𝑧)2||||+101𝜌Re𝜌𝑧(1+𝜌𝑧)(1𝜌𝑧)2𝑑𝜌=||||𝑧log(1𝑧)2||||exp101𝜌Re𝜌𝑧(1+𝜌𝑧)(1𝜌𝑧)2𝑑𝜌=||||𝑧log||||(1𝑧)expRe2𝑧1𝑧.(2.20)
Therefore, (log|𝐻(𝑧)|)||||𝑧log||||(1𝑧)expRe2𝑧1𝑧,(2.21) which is part (a).
By (2.4), (2.15) and in similar fashion to the upper part, (log|𝑔(𝑧)|)101(𝜌)Re𝜌𝑧(1+𝜌𝑧)(1𝜌𝑧)2=||𝑑𝜌log1𝑧||expRe2𝑧1𝑧.(2.22) (2.21) and (2.22) give part (c).

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

Corollary 2.3. If 𝑓(𝑧)=𝑧(𝑧)𝑔(𝑧) is starlike Log-harmonic, then, for 𝑧=𝑒𝑖𝜃, ||||𝑓(𝑧)𝑝||||𝑑𝑡𝑧|expRe4𝑧|||1𝑧𝑝𝑑𝜃,𝑝>0,log+||||𝑓(𝑧)𝑑𝑡log+|𝑧|𝑑𝜃+Re4𝑧1𝑧𝑑𝜃𝐶<,(2.23) the later implies that 𝑓𝑁+ 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.

Theorem 2.4. If 𝑓(𝑧)=𝑧(𝑧)𝑔(𝑧)STLh and for 𝑟,0<𝑟<1,|𝑓(𝑟𝑒𝑖𝜃)|𝑀(𝑟), then 𝐿(𝑟)4𝜋𝑀(𝑟)/(1𝑟2).

Proof. Let 𝐶𝑟 denote the closed curve which is the image of the circle |𝑧|=𝑟<1 under the mapping 𝑤=𝑓(𝑧). Then 𝐿(𝑟)=𝐶𝑟||||=𝑑𝑓02𝜋||𝑧𝑓𝑧𝑧𝑓𝑧||=𝑑𝜃02𝜋||𝑓||||||𝑧𝑓𝑧𝑧𝑓𝑧𝑓||||𝑑𝜃𝑀(𝑟)02𝜋||||𝑧𝑓𝑧𝑧𝑓𝑧𝑓||||𝑑𝜃.(2.24) Now using (2.1), we have 𝑧𝑓𝑧𝑧𝑓𝑧𝑓=Re𝑧𝜑𝜑+𝑖Im1+𝑎1𝑎𝑧𝜑𝜑.(2.25)
Therefore, 𝐿(𝑟)𝑀(𝑟)02𝜋Re𝑧𝜑𝜑𝑑𝜃+𝑀(𝑟)02𝜋||||Im1+𝑎1𝑎𝑧𝜑𝜑||||𝑑𝜃=𝑀(𝑟)𝐼1+𝑀(𝑟)𝐼2.(2.26)
Since Re(𝑧𝜑/𝜑) is harmonic, and by the mean value theorem for harmonic functions, 𝐼1=2𝜋. Moreover, ((1+𝑎)/(1𝑎))(𝑧𝜑/𝜑) is subordinate to ((1+𝑧)/(1𝑧))2; therefore, we have 𝐼202𝜋|||1+𝑧|||1𝑧2𝑑𝜃=2𝜋1+2𝑛=1𝑟2𝑛=2𝜋1+𝑟21𝑟2.(2.27)
Substituting the bounds for 𝐼1 and 𝐼2 in (2.26), we get 𝐿(𝑟)2𝜋𝑀(𝑟)+2𝜋𝑀(𝑟)1+𝑟21𝑟24𝜋𝑀(𝑟)1𝑟2.(2.28)

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