A New Look for Starlike Logharmonic Mappings

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 , z z f f a f f = where ( ) a H U ∈ such that ( ) a U U ⊂ . These mappings admit a global representation of the form ( ) ( ) ( ) 2 , β f z z z h z g z = where ( ) . 1 2 β R > − In this paper,we shall consider the logharmonic mappings ( ) ( ) ( ) f z zh z g z = , where ( ) zh z = φ 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.

where m is a nonnegative integer, and, h and g are analytic functions in U with ( ) and ( ) 0 0 h ≠ ( [1]).The exponent β in (1.2) depends only on ( ) and can be expressed by and that a univalent logharmonic mapping on U vanish at the origin if and only if 1 m = .Thus, a univalent logharmonic mappings on U which vanishes at the origin will be of the form and have been studied extensively in the recent years, see [1]- [7].In this case, it follows that ( ) ( ) 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 2 f z z hg β = be a univalent logharmonic mapping.We say that f is starlike logharmonic mapping if ( )

Denote by
Lh ST * the set of all starlike logharmonic mappings, and by S * the set of all starlike analytic mappings.It was shown in [4] that It is rather a natural question to ask whether there exists a linkage between the starlikeness of ( ) ( ) ( ) f z zh z g z = and ( ) ( ).
In Section 2, we determine the radius of starlikeness for the logharmonic mapping ( ) 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.

Basic Properties of Mappings from
.

Lh S *
We start this section by establishing a linkage between the starlikeness of ( ) Then ( ) This gives ( ) Therefore, the radius of starlikeness ρ is the smallest positive root (less than 1) of f z zhg = be a starlike logharmonic mapping defined on the unit disk U with respect to a B ∈ with ( ) and also, and then simple calculations give that Therefore, the radius of starlikeness ρ is the smallest positive root (less than 1) of < onto a starlike domain.□ Our next result is a distortion theorem for the set of all logharmonic mappings ( ) be a logharmonic mapping defined on the unit disk U where Proof.i) Let ( ) be a logharmonic mapping with respect to a B ∈ with ( ) Then ( ) 1 Combining (2.2), (2.3) and (2.4), we get ii) and iii) Differentiation ( ) 2) with respect to z and z respectively leads to 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 ( ) be a logharmonic mapping defined on the unit disk U where ( ) ( ) .
Proof.Let r C denote the closed curve which is the image of the circle 1 z r = < under the mapping ( ) Now using (2.5) and (2.6) we have is harmonic, and by the mean value theorem for harmonic functions, 1 2π.
Substituting the bounds for 1 I and 2 I in (2.8), we get

Integral Means
Theorem 4 of this section is an applications of the Baerstein star functions to the class of logharmonic mappings ( ) where ( ) ( ) e sup e d , for .
i it One important property is that when u is symmetric (even) re-arrangement then 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 , g h real and 1

L on [ ]
, , a a − the following are equivalent a) For every convex non-decreasing function a a a a g x x h x x a a a a g x t t h x t t 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, Let B denote the set of all analytic functions a defined on the unit disk defined on the unit disk U is a solution of the nonlinear elliptic partial differential equation , hence, non-constant logharmonic mappings are sense-preserving and open on U. If f is a nonconstant logharmonic mapping of U and vanishes at 0 z = but has no other zeros in U, then f admits the following representation Equality holds for the right hand side if and only if ( )

∫=
Equality holds for the right side if and only if ( ) For the left hand side inequality, we have

Theorem 4
g x h x * * ≤Our main result of this section is the following theorem.If ( )f z zhg =be a logharmonic mapping defined on the unit disk U where ( ) ( ) mapping defined on the unit disk U where ( -decreasing convex function .The first integral mean can be obtained using part (a) of Lemma 1 and Theorem 4.Moreover, the choice