SUBCLASSES OF UNIVALENT FUNCTIONS SUBORDINATE TO CONVEX FUNCTIONS

In this paper, we define a new subclass Ao(A, B) of univalent functions and investigate several interesting characterization theorems involving a general class S" [A, B] of starlike functions


INTRODUCTION AND DEFINITIONS
Let .A denote the class of functions normalized by f(z) z + anz n, (I I) which are analytic in the open unit disk ( { z z E C and Izl < I }.Further, let S denote the class of all functions in L/which are univalent in L/ A function f(z) belonging to ,.q is said to be starlike of order a (0 _< a < 1) if and only if We denote by S* (a) the subclass of S consisting of functions which are starlike of order a A function f(z) belonging to 8 is said to be convex of order a (0 < a < 1) if and only if z.f."(z) We denote by/C(a) the subclass of 8 consisting of functions which are convex of order a We note that S*(a) C_ S* (O) _= S* (O_<o< I) and c() c_ c(0) c (0 _< < 1).
( ) With a view to introducing an interesting family of analytic functions, we should recall the concept of subordination between analytic functions Given two functions f(z) and g(z), which are analytic in L, the function f(z) is said to be subordinate to g(z) if there exists a function h(z), analytic in L/with (0) 0 and I() such that () (()) ( s u).

MAIN RESULTS
Applying the method of the integral representation [2] for functions in .Mo(A, B) (a > 0), it is not difficult to deduce LEMMA 1.The function f(z) is in JVlo(A,B), a > 0, if and only if there exists a function g(z) belonging to the class S" [A, B] such that f(z) {g(t)}l/t-ldt (2 1) PROOF.Setting g(z) f(z) {zf'(z)/f(z)} , so that (2.1) is satisfied, we observe that zg'(z) zft(z) Hence f e Jvlo (A,B) ifand only ifg S'[A,B].
Before stating our first theorem, we need the following definition DEFINITION 2. Let c be a complex number such that Re c > 0, and let If h is the univalent function h(z) 2Nz/(1-z2) and b h-l(c), then we define the "open door" (ef [3]) function Q as Qc(z) hI(z + b)/(l +-z)], z E L. (23) TItEOREM 1.Let f E .h4 (A, B) (a > 0), and let l+bz -a2_(z). ( 24 Then f S" PROOF.Since f .Mo (A, B) (a > 0), it follows that there exists a function g S*[A, B] such that 1 {g(t)}l/'t-ldt I(*) by using Lemma 1.By the hypothesis, we also have Thus, by a result ofMiller and Mocanu ([3], Corollary 3.1), we have f(z) {g(t)}l/t-ldt e S'.

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: