On the Fourth Coefficient of Bounded Univalent Functions

Abstract

This class has often been studied in the equivalent form of normalized univalent functions F{z) = fe1_1/(2) = z+ Ï tv\ (2) a, = A, \Fiz)\ib;K We shall study estimates for the coefficient aA in dependence on the first coefficient by .¡For this purpose an inequality of the Grunsky-Nehari type [2], [3] will be used. Such inequalities were first applied by V. Singh [4] in studying the coefficient a4 of bounded univalent functions with real coefficients. The method of Singh, which follows the argument of Charzynski and Schiffer [1] in the case of unbounded univalent functions, can be extended to the case of nonreal coefficients. Though much harder to apply here than in the real case, this method yields sharp bounds for a4 in an interval of ¿^-values near by = 0 and in an interval near by = 1. 2. We consider the univalent and bounded function (3) F(z) = V/(z2) and apply to it the Grunsky-Nehari inequalities for such functions. We use specifically the estimate (4) 13 / 3 \ a4-2a2a3 +-a\ + x\a2 + x2(a3-a2) + 2xt la3-a\ I Ú |(1-b\)-\by\a2\2 + 2|x1|2(l-by) + |x2|2(l-b2y)-2byRe{xyä2}.

Classical and Stochastic Löwner–Kufarev Equations

“…The Pick function does not give the maximum to |a 3 | and the estimate is much more difficult. Schiffer and Tammi [229] in 1965 found that |a 4 | ≤ p 4 (M) for any f ∈ S M with M > 300. This result was repeated by Tammi [247, page 210] in a weaker form (M > 700) and there it was conjectured that this constant could be decreased until 11.…”

Coefficient bounds for some classes of starlike functions

“…Hence the above inequality is also sharp for functions in 5(0 when 1< t ^ δ π . Finally we remark that Schiffer and Tammi [6] have shown that if suffices to take δ 4^ 34/19. We now use Theorems A, B, and C to prove Theorem 1.…”

“…If f(z) = z + Σ x n=2 a n z\ z E K, is in l/(O,then Schiffer and Tammi [6] showed that \a 4 \^Λ 4 (t), for ί ^ 33 1/3. If in addition / has real coefficients, then Singh [8] proved that \a 4 \^A 4 (t) for t ^ 11.…”

Optimal Control in Bombieri's and Tammi's Conjectures

“…The Pick function does not give the maximum to |a 3 | and the estimate is much more difficult. M. Schiffer and O. Tammi [15] in 1965 found that |a 4 | ≤ p 4 (M ) for any f ∈ S M with M > 300. This result was repeated by O. Tammi [18, page 210] in a weaker form (M > 700) and there it was conjectured that this constant could be decreased until 11.…”