Optimal Control in Bombieri's and Tammi's Conjectures

Abstract

Let 𝑆 stand for the usual class of univalent regular functions in the unit disk 𝑈 = {𝑧 : |𝑧| < 1} normalized by 𝑓(𝑧) = 𝑧 + 𝑎2𝑧2 + . . . in 𝑈, and let 𝑆𝑀 be its subclass defined by restricting |𝑓(𝑧)| < 𝑀 in 𝑈, 𝑀 ≥ 1. We consider two classical problems: Bombieri's coefficient problem for the class 𝑆 and the sharp estimate of the fourth coefficient of a function from 𝑆𝑀. Using Löwner's parametric representation and the optimal control method we propose a way to finding initial Bombieri's numbers and derive a sharp constant 𝑀0 such that for all 𝑀 ≥ 𝑀0 the Pick function gives the local maximum to |𝑎4|. Numerical approximation is given.

Classical and Stochastic Löwner–Kufarev Equations

“…It is easily seen that σ 43 = B 43 = σ 23 = B 23 = 0. Applying Löwner's parametric representation for univalent functions and the optimal control method we found [209] the exact Bombieri numbers σ 42 , σ 24 , σ 34 and their numerical approximations σ 42 ≈ 0.050057 . .…”

“…The case of function with real coefficients is simpler: the Pick function gives the maximum to |a 4 | for M ≥ 11 and this constant is sharp (see [246], [248, page 163]). By our suggested method we showed [209] that the Pick function locally maximizes |a 4 | on S M if M > M 0 = 22.9569 . .…”

On the failure of Bombieri's conjecture for univalent functions

“…Note that the polynomial (13) resembles the polynomials considered in a theorem of Brannan [4,Thm. 3.1], which gave necessary and sufficient conditions for a polynomial of the form z + a z n + z 2n−1 2n − 1 , a ∈ C, to be univalent.…”

On the Failure of Bombieri’s Conjecture for Univalent Functions

“…Proofs (disproving the conjecture) for the points (2, 4), (3,4) and (4, 2) were then furnished by Prokhorov and Vasil'ev [13], who computed (approximately) the corresponding Bombieri numbers. Recently, Leung [10] developed a variational method which allowed him to show that σ m2 < B m2 for all m ≥ 3 and that σ m3 < B m3 for all odd m ≥ 5.…”