ON THE ASYMPTOTIC BIEBERBACH CONJECTURE

The set S consists of complex functions f, univalent in the open unit disk, with f(0) f' (0) 1 0. We use the asymptotic behavior of the positive semidefinite FitzGerald matrix to show that there is an absolute constant N such o n that, for any f(z) z + an z S with la31 N o

Let S denote the class of all normalized univalent functions f(z) z + a k z in the open unit disc D. The Bieberbach conjecture states that, for k=2 functions in S, one has anl _< n for all n N. It is known to be true for n <-6.
The best known estimate for all coefficients is la -< (I.066)n (Horowitz [i]).On the other hand, Hayman's Regularity Theorem (Hayman [2]) states that lira _< 1 n n-o z for each f e S and that equality holds only for the Koebe function K(z) 2' (l-nz) JNI i, for which JanJ n.This implies that lanl -< n for n -> no(f).
Hayman [3] also proved that A /n tends to a limit, where A is the maximum of n n anl for all f S. It is still an open question as to whether this limit is equal to i.The asymptotic Bieberbach conjecture asserts that lim A /n 1, where n An max ..lanl-feS Ehrig [4] has proved via the FitzGerald Inequality [5] that if f S and laB1 < C < 2.43, then lanl < n for all n _> No, where No depends only on C and not (as in Hayman's Regularity Theorem) on f.This result is a proof of the Asymptotic Bieberbach Conjecture for a subclass of S.
In this paper, we apply the Asymptotic FitzGerald Inequalities to get, by ele- mentary means, an improvement of Ehrig's result (Theorem i) and the result in [6], (see Remark 2).
THEOREM A. (FitzGerald Inequality, [5] .Let where bj(f) .laj(f)I; j(m,n) j(n,m), j N, and for m < n: m-lj-n for lj-nl < m j (m,n) Then the FitzGerald matrix THEOREM B. (Asymptotic FitzGerald Inequalities [7]).Let {f n e N, be a n sequence of functions in S, such that a) f converges locally uniformly to f 'Jr-l' (f) ,(j),8,d d) (f), defined below, is a positive semi-deflnite matrix.
Denote by E the m x n matrix whose elements are all equal to one.Moreover, let H (f) be the m n matrix defined by its elements hst(f j2 b 2 (f).We suPk bn(fk)/n.Then matrix A has the following form: n n n-o (f)HTr_ I q r(f), Mq r(e(f)), 82(l-e2(f))Eq_ r i d2(l-2(f))Eq r,p-q 2HTr-l,l(f)' B2(I 2(f))El, q r' (762/6 B4)EI,I d2(I B2)El,p_q where H T is the transposed matrix of H.

MAIN RESULTS.
For the proof of the Theorem i, we need the following lemmas: LEMMA i. Suppose that n > i and that (H) where H is a compact subclass of S. Let f(z) Here, a (v) are the coefficients of f(z) v, where The proof of this lemma is completely similar to that of Theorem i in Schaeffer and Spencer ([8], p. 612).
As an application of the Lemma i, we have the following: a z e H is the extremal function maximizing n n=2 such that a 3 > 0, then 2a 3 a + 2.
The proof of this lemma is completely similar to that in Garabedian and   Schiffer ([9], p. 118).
LEMMA 5. [7].Let {f }, n N, be a sequence of univalent functions in S, n that converges locally uniformly to a function f in S and suppose that (f) > O.
Then 762 2(f) >_ 64 where and 6 are chosen as in theorem B PROOF.Consider the (q r + l) Mq_r ( (f) B2 (i_2 (f))Eq_r, 2 (i-2 (f))El,q-r (762/6-4) El i of the matrix A in theorem B. A well-known result about positive semidefinite quadratic form is that all principal minor determinants of the matrix of the coef- ficients of the quadratic form are non-negative.Let (f) and n q-r.If we use induction, we obtain: Det Q( ,) (2/6)n (i_2) (762_64/2 + 2n 6-(n+l) (72_6B4/2) O; hence, for 0 -< < I (762 2 64) + 6(I-2)B 4 > 0o 6n (i-2) +i Since n is arbitrary, the resultlows.The case i is immediate.k THEOREM I. Let f(z) z + a k z be in S. If i < la31 < 2.58, then there is an absolute constant N (independent of f), such that la < n for all n > N o n o PROOF.Suppose the contrary and take a sequence {gk }, k e N, of univalent functions in S such that i) {gk }, k e N, converges locally uniformly to a function go S, ii) 1 < b3(gk a3(gk <-2.58, 2 iii) 2a 3(gk a 2(gk + 2, iv) bnk(g k) -> n k for sequence n k going to infinity.
REMARK.The functions are the extremal functions maximizing la3(gk) in the compact subclass H k {g S; i -< b3(g) < 2.58 and bnk(g) > n k} of S. Applying Lemma 2 to the subclass Hk, we obtain condition (iii).
We pick for each n k one of the functions of {gj }, j 0,I which maximizes bnk and denote it by fnk; precisely, let {fnk }, k N, be a sequence such that {gj}, j 0,i sup b (fnk).j nk (gj bnk We may assume that {f }, k N, converges locally uniformly to a function f S.
n k Otherwise, we pick a subsequence of {f }, k e N. Evidently, i -< b3(f) < 2.58 and n k 2 2a3(f a2(f) + 2. For this sequence {f }, k N, we have n k nk sp fn )/nk fn k) bnk j bnk /n k Thus lim sup b (fnk)/n k > i.
This implies, by Lemma 4, that b3(f) > 2.58 which contradicts the assumptions.k COROLLARY.Let f(z) z + =a z be in S. If laB1 < 2.58, then there is k=2 an absolute constant N (independent of f), such that la < n for all n > N o n o PROOF.The proof of corollary follows immediately from Theorem i and Theorem C.
ACKNOWLEDGEMENT.This research was supported in part by FINEP and CNPQ.

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: Moo(r,f is the maximum of If(z) on Izl r.The number (0 -< -< I) is called the Hayman index of f. n LEMMA3.[i0].If f(z)

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