REMARK ON FUNCTIONS WITH ALL DERIVATIVES UNIVALENT

An attractive conjecture is discounted for the class of normalized uni- valent functions whose derivatives are also univalent.


f(z)
z + L akz k=2 (i) which are analytic and univalent in the unit disk D: zl < i. Denote by E the set of functions f(z) in S for which the n-th derivative f(n)(z) is univalent in D for each n i, 2, 3, Next we set '= sup{la21" f(z) e E}. (2) It is known (cf.[2], [3]) that /2 < e < 1.7208, the left hand inequality being a consequence of the fact that (e z i)/ e E.
It was the belief of several authors that the function (e z I)/ was extremal for (2).The purpose of the present note is to show that n/2 is not sharp in the inequality (3).This is accomplished by exhibiting other members of the set E which improve the lower estimate on e in (3).
In particular, we consider perturbations of e z of the form e WZ F(z) F(z; a,b): + a(z + bz2). (4) for real parameters a > 0 and b > /2.For certain values of a and b, the analytic function (F(z; a,b) i)/( + a) can be shown to be in E, which yields the estimate _ + a(b /2) < a.
(5) (6) Consequently we obtain the following improvement in (3), /2 + .02< 1.5910 (7) Before discussing the proof of the above proposition, we motivate the particular choice of parameters a and b by examining what restrictions are imposed upon a and b by the univalence of F(z; a,b) defined in (4).
As F'(z; a,b) cannot vanish for z in D, and, in particular, for z in (-I,i), it follows that -/ 0 < a < e (2b I) Furthermore, since F(; a,b) F(z; a,b), the imaginary part of F(z; a,b) must remain positive for all z in D with Im z > 0. After some simple computations, this last remark implies, for b fixed and x g (-i, -I/2b), that -e sin x a < (9 Hence to optimize the lower bound on provided by (5) we select a e /(2b I) and choose b as large as possible so that the inequality of (9) remains valid.This maximal value appears to be near 18.9851.However, as little improvement is gained in (7) by this extreme choice for b, we make the more convenient special choice of b 18 and a e /35 in the Proposition.
PROOF OF PROPOSITION We now return to the univalence question for F(z; e /35, 18) and its derivatives.From the definition of F(z; ab) in (4), for IF(z; e-/35, 18) I]/[ + e-/35] to be in E, we need only show that F(z; ge-g/35, 18) and its first derivative are univalent in D.Moreover, it suffices to show that each of these functions is univalent on the unit circle Izl i (cf. [i]).As the proof is rather technical, we only sketch the -/ procedure for the univalence of F(z; e 35, 18).
It can be verified (cf. Figure I) that u(e) is strictly decreasing on (0,e 2) and strictly increasing on (e2,) for some e univalence of F(z; a,b) on the unit circle and hence in D, for the particular choices of a and b stated in the Proposition.Similarly, we may establish the univalence of F'(z; a,b).The Figure 1 is slightly exaggerated to demonstrate the behavior of v(@) near the real line.
((), v()) for calling this problem to my attention, and secondly for their helpful remarks and suggestions.Also l'd like to thank Prof. S. M. Shah for his kind interest and comments.This work was done while at the University of South Florida, Tampa, Florida.

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:

First
Round of Reviews March 1, 2009