© Hindawi Publishing Corp. AN APPLICATION OF A SUBORDINATION CHAIN

Let K denote the class of functions g ( z ) = z + a 2 z 2 + ⋯ which are regular and univalently convex in the unit disc E . In the present note, we prove that if f is regular in E , f ( 0 ) = 0 , then for g ∈ K , f ( z ) + α z f ′ ( z ) ≺ g ( z ) + α z g ′ ( z ) in E implies that f ( z ) ≺ g ( z ) in E , where 0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"> α > 0 is a real number and the symbol  ≺  stands for 
subordination.


Introduction. Let S denote the class of functions
which are regular and univalent in the unit disc E = {z : |z| < 1}.A function f ∈ S is said to be convex of order β, 0 ≤ β < 1, if and only if For a given β, 0 ≤ β < 1, let K(β) denote the subclass of S consisting of convex functions of order β and let K = K(0) be the usual class of convex functions.A function f given by (1.1) is said to be close-to-convex in E if f is regular in E and if there exists a function g ∈ K such that Re f (z) g (z) > 0, z ∈ E. (1. 3) Suppose that f and g are regular in |z| < ρ and f (0) = g(0).In addition, suppose that g is also univalent in |z| < ρ.We say that f is subordinate to g In 1947, Robinson [4] proved that if g(z)+zg (z) is in S and f (z)+zf (z) ≺ g(z) + zg (z) in |z| < 1, then f (z) ≺ g(z) at least in |z| < r 0 = 1/5.S. Singh and R. Singh [6], in 1981, increased the constant r 0 to 2 − √ 3 = 0.268 .... Subsequently, in 1984, Miller et al. [2] further increased this constant to 4− √ 13 = 0.3944 .... Recently, R. Singh and S. Singh [5] pursued the problem initiated by Robinson when g ∈ K(β).In fact, they considered the cases when β = 0 and β = 1/2 and proved the following results.

Preliminaries.
We will need the following definition and results to prove our theorem.
Lemma 2.3.Let p be analytic in E and q analytic and univalent in E except for points where lim z→ς p(z) = ∞ with p(0) = q(0).If p is not subordinate to q, then there is a point Lemma 2.3 is due to Miller and Mocanu [1].

Main theorem
Theorem 3.1.Let f be regular in E with f (0) = 0 and let g ∈ K.For any real number α, α > 0, suppose that Proof.First, we observe that g(z)+αzg (z) = h(z), say, is close-to-convex and hence univalent in E whenever g ∈ K. Without any loss of generality, we can assume that g is regular and univalent in the closed disc E. If possible, suppose that f (z) is not subordinate to g(z) whenever (3.1) holds.Then by Lemma 2.3, there exist points ( Define a function Since h(z) and zg (z) are analytic in E, L(z, t) is also analytic in E for all t ≥ 0, and is continuously differentiable on [0, ∞) for all z ∈ E. Now, from (3.4), we get for all t ≥ 0 and α > 0. Also where z 0 ∈ E, |ζ 0 | = 1, and m ≥ 1. Formula (3.9), when combined with (3.8), contradicts (3.1).Hence, we must have f (z) ≺ g(z) in E. This completes the proof of our theorem.
Letting α approach infinity, we arrive at the following well-known result of Suffridge [7].
We now present some interesting examples choosing g as some distinguished member of the class K.

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: