A NOTE ON LOGHARMONIC MAPPINGS

where (a) m is 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 U , g(0)= 1, and h(0)≠ 0. Univalent logharmonic mappings on the unit disc have been studied extensively. For details see [1, 2, 3, 4, 5, 6, 7, 8]. Suppose that f is a univalent logharmonic mapping defined on the unit disc U . Then, if f(0) = 0, the function F(ζ) = log(f (eζ)) is univalent and harmonic on the half plane {ζ : Reζ < 0}. For more details on univalent harmonic mappings defined in the unit disc U , see [9, 10, 11, 12]. In this note, we consider the problem of minimizing the moments of order p over a subclass of the class logharmonic mappings defined over the unit disc U . It is interesting to note that the extremal functions are univalent starlike logharmonic mappings.


Introduction.
Let H(U) be the linear space of all analytic functions defined on the unit disc U = {z = x + iy : |z| < 1}.A logharmonic mapping is a solution of the nonlinear elliptic partial differential equation where the second delatation function a is in H(U) and |a(z)| < 1 for all z ∈ U .If f does not vanish on U, then f is of the form where H and G are in H(U).On the other hand, if f vanishes at 0 but has no other zeros in U , then f admits the representation and therefore, β > −1/2, (c) h and g are analytic in U , g(0) = 1, and h(0) ≠ 0. Univalent logharmonic mappings on the unit disc have been studied extensively.For details see [1,2,3,4,5,6,7,8].Suppose that f is a univalent logharmonic mapping defined on the unit disc U.Then, if f (0) = 0, the function ) is univalent and harmonic on the half plane {ζ : Reζ < 0}.For more details on univalent harmonic mappings defined in the unit disc U, see [9,10,11,12].
In this note, we consider the problem of minimizing the moments of order p over a subclass of the class logharmonic mappings defined over the unit disc U .It is interesting to note that the extremal functions are univalent starlike logharmonic mappings.

Moments of order p
Theorem 2.1.Let f = zh(z)g(z) be logharmonic mapping defined on the unit disc U such that h(0) = g(0) = 1.Let M p (r , f ) denote the moment of order p, p ≥ 0.Then, ZAYID ABDULHADI

Equality holds if and only if
or one of its rotations ηf 1 (ηz).
Remark 2.2.If p = 0 in Theorem 2.1, then we have the problem of minimizing the area.Moreover, if p = 2, then we obtain the minimum of the moment of inertia.
Proof.Let f = zh(z)g(z) be logharmonic mapping defined on the unit disc U.Then, f satisfies (1.1) for some a ∈ H(U) such that |a(z)| < 1 and a(0) = 0. Hence, using Schwarz's lemma, we have (2.3) and therefore, Equality holds if and only if and a(z) = ηz, |η| = 1.This implies that and then, The solution of the differential equation

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: