© Hindawi Publishing Corp. A FACTORIZATION THEOREM FOR LOGHARMONIC MAPPINGS

We give the necessary and sufficient condition on 
sense-preserving logharmonic mapping in order to be factorized as 
the composition of analytic function followed by a univalent 
logharmonic mapping.

Let D be a domain of C and denote by H(D) the linear space of all analytic functions defined on D. A logharmonic mapping is a solution of the nonlinear elliptic partial differential equation where a ∈ H(D) and |a(z)| < 1 for all z ∈ D. If f does not vanish on D, then f is of the form where H and G are locally analytic (possibly multivalued) functions on D. On the other hand, if f vanishes at z 0 , but is not identically zero, then f admits the local represenation where (a) m is a nonnegative integer, (b) β = a(0)(1 + a(0))/(1 −|a(0)| 2 ) and therefore β > −1/2, (c) h and g are analytic in a neighbourhood of z 0 .In particular, if D is a simply connected domain, then f admits a global representation of the form (3) (see, e.g., [2]).Univalent logharmonic mappings defined on the unit disk U have been studied extensively (for details, see, e.g., [1,2,3,4,5,6]).
In the theory of quasiconformal mappings, it is proved that for any measurable function µ with |µ| < 1, the solution of Beltrami equation f z = µf z can be factorized in the form f = ψ • F, where F is a univalent quasiconformal mapping and ψ is an analytic function (see [8]).Moreover, for sense-preserving harmonic mappings, the answer was negative.In [7], Duren and Hengartner gave a necessary and sufficient condition on sense-preserving harmonic mapping f for the existence of such factorization.Since logharmonic mappings are preserved under precomposition with analytic functions, it is a natural question to ask whether every sense-preserving logharmonic mapping can be factorized in the form f = F • φ for some univalent logharmonic mapping F and some analytic function φ.
It is instructive to begin with two simple examples.
Example 1.Let f be the logharmonic mapping f (z) = z 2 /|1 − z| 4 defined on the unit disc U .Then f is sense-preserving in U with dilatation a(z) = z.We claim that f has no decomposition of the desired form in any neighborhood of the origin.Suppose on the contrary that f = F • φ, where φ is analytic near the origin and F is univalent logharmonic mapping on the range of φ.Then F is sense-preserving because f is.Without loss of generality, we suppose that , where H and G are analytic and have power series expansion where 2 , the function φ must have an expansion of the form It follows that G • φ has an expansion of the form Hence, f has no factorization of the form f = F • φ of the required form in any neighborhood of the origin.
Now, we state and prove the factorization theorem.Theorem 3. Let f be a nonconstant logharmonic mapping defined on a domain D ⊂ C and let a be its dilatation function.Then, f can be factorized in the form f = F • φ, for some analytic function φ and some univalent logharmonic mapping F if and only if Under these conditions, the representation is unique up to a conformal mapping; any other represenation f = F 1 • φ 1 has the form F 1 = F • ψ −1 and φ 1 = ψ • φ for some conformal mapping defined on φ(D).
Proof.Suppose that f = F • φ, where F is a univalent logharmonic mapping and φ is an analytic function.Let A(ζ) be the dilatation function of F .Then simple calculations give that f z = F w (φ)φ , f z = F w (φ)φ , and a(z) = A(φ(z)).Since F is univalent, the Jacobian is nonzero and hence |a(z)| = |A(φ(z))| ≠ 1 (see [2]).Also, F is univalent and f (z Next, suppose that the two conditions are satisfied.We want to show that f can be factorized in the form f = F(φ).This is equivalent to finding a univalent continuous function G defined on f (D) so that G •f is analytic.In view of the Cauchy-Riemann conditions, this is equivalent to where ), and a(z 1 ) = a(z 2 ), it follows that b(z 1 ) = b(z 2 ).Hence, µ(w) is well defined and |µ(w)| ≠ 1 for all w ∈ f (D).
Let {D n } be an exhaustion of D, Ω n = f (D n ) and let µ n be the restriction of µ to Ω n .Extend µ n to C by assuming that where w 0 ,w 1 ∈ Ω 1 and w 1 ≠ w 0 .This is possible because f is not constant on D 1 .Then, H n is also a homeomorphic solution to the Beltrami equation, normalized to satisfy H n (w 0 ) = 0, H n (w 1 ) = 1, and H n (∞) = ∞.This and the fact that each H n is K-quasiconformal mapping on Ω j imply that H n converges locally uniformly to a K-quasiconformal mapping H on Ω j .It follows that H is a homeomorphism on Ω and H satisfies the equation Hence Next, we show that F = H −1 is logharmonic mapping.Note that f = F • φ was assumed to be logharmonic in D.Then, near any point ζ = φ(z) where φ (z) ≠ 0, we can then deduce that F = f • φ −1 is logharmonic, where φ −1 is a local inverse.But F is locally bounded, so the (isolated) images of critical points of φ are removable, and F is logharmonic mapping on φ(D).
Finally, we prove the uniqueness.Suppose that

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: and analytic, and G 0w = µG 0w .But the solution of this Beltrami equation is unique; hence, G 0 = G.This completes the proof of the theorem.