Exact determination of the Bombieri numbers $\sigma_{mn}$: the limit inferior of $(n-\mathrm{Re}\,a_n)/(m-\mathrm{Re}\,a_m)$ at the Koebe function

For the class $S$ of normalized univalent functions in the unit disk and the Koebe function $K(z)=z/(1-z)^2$, Bombieri (1967) introduced the numbers $\sigma_{mn}=\liminf_{f\to K,\,f\in S}\frac{n-\mathrm{Re}\,a_n}{m-\mathrm{Re}\,a_m}$ ($m,n\ge2$) while proving the local maximum property of $K$ for odd coefficients, and conjectured (his equation (5.1)) that $\sigma_{mn}=B_{mn}:=\min_\theta\frac{n\sin\theta-\sin n\theta}{m\sin\theta-\sin m\theta}$ for all $m,n\ge2$. The conjecture is now known to be false in broad classes of cases (first exactly at $(m,n)=(3,2)$, where Greiner and Roth computed $\sigma_{32}=(e-1)/4e<1/4=B_{32}$), trivially true when $m$ is even and $n$ odd, and true for functions with real coefficients; but it remains undecided for most pairs with $m<n$ and for pairs with $m$ odd, $n$ even, $n/m$ in the critical region above $0.8194$, and the underlying source problem — the exact value of the limit inferior — is open for every pair except $(3,2)$ and trivial zero cases, with only bounds $0\le\sigma_{mn}\le B_{mn}$ and a few optimal-control numerical values known. The problem is to determine the Bombieri numbers $\sigma_{mn}$ exactly and to decide the conjecture in the remaining undecided cases.

Background

A holomorphic function on the unit disk is called univalent (or schlicht) if it is one-to-one; $S$ denotes the class of normalized univalent functions $f(z)=z+a_2z^2+a_3z^3+\cdots$. The Bieberbach conjecture (1916), proved by de Branges in 1985, states that $|a_n|\le n$ for $f\in S$, with equality only for rotations of the Koebe function $K(z)=z/(1-z)^2=z+2z^2+3z^3+\cdots$, whose coefficients are all real and maximal. The coefficient body $V_n=\{(a_2,\ldots,a_n):f\in S\}$ therefore has $K$'s coefficient point $(2,3,\ldots,n)$ as a boundary corner. Bombieri's 1967 paper belongs to the local theory at this corner: the local maximum property states that functions in $S$ sufficiently close to $K$ (uniformly on compacta) still satisfy $|a_n|\le n$; its even-coefficient case was obtained from the Grunsky inequalities (a classical system of coefficient conditions implied by univalence) by Garabedian, Ross and Schiffer (1965), the odd case in the same normalization by Garabedian and Schiffer (1967). For odd $n$, however, the deficit $n-\mathrm{Re}\,a_n$ can vanish faster than linearly in $2-\mathrm{Re}\,a_2$ — it is bounded below only by a constant multiple of $(2-\mathrm{Re}\,a_2)^{3/2}$ near $K$ — so the $a_2$-denominated ratio degenerates, and Bombieri's sharp quantitative form for odd $n$ normalizes with the denominator $3-\mathrm{Re}\,a_3$ instead. Quantifying this local geometry, Bombieri introduced what are now called the Bombieri numbers $\sigma_{mn}$: the limit inferior of the ratio of coefficient deficits $\frac{n-\mathrm{Re}\,a_n}{m-\mathrm{Re}\,a_m}$ as $f\in S$ tends to $K$, describing the local shape of the coefficient body at the Koebe corner. The conjectured value $B_{mn}$ is the minimum of a trigonometric ratio suggested by the two-slit (Pick) maps $z/\bigl((1-e^{i\theta}z)(1-e^{-i\theta}z)\bigr)$, whose coefficients equal $\sin(k\theta)/\sin\theta$ for $k\ge1$; note $B_{mn}=0$ exactly when $m$ is even and $n$ odd, where the conjecture is trivially true. The principal modern tool is Loewner's parametric representation (1923): a dense subclass of $S$ consists of the functions $f(z)=\lim_{t\to1^-}w(z,t)/(1-t)$, where $w$ solves the linear Loewner ODE $\frac{dw}{dt}=-\frac{w}{1-t}\cdot\frac{e^{iu(t)}+w}{e^{iu(t)}-w}$, $w|_{t=0}=z$, with a piecewise continuous real control function $u$ (the constant control $u\equiv\pi$ produces $K$); through it $\sigma_{mn}$ becomes an optimal-control problem (Pontryagin maximum principle, singular controls near the Koebe point), and second-variation computations along explicit control families give upper bounds for $\sigma_{mn}$. Support points of $S$ — exposed points of the coefficient body for linear functionals such as $\mathrm{Re}(a_3+\lambda a_2)$ — yielded the first exact evaluation.

Problem Statement

For integers $m,n\ge2$ with $m\ne n$, determine the exact value of the Bombieri number $\sigma_{mn}=\liminf_{\substack{f\to K\\ f\in S}}\frac{n-\mathrm{Re}\,a_n}{m-\mathrm{Re}\,a_m},$ where $S$ is the class of normalized univalent functions $f(z)=z+\sum_{k\ge2}a_kz^k$ in the unit disk, $K(z)=z/(1-z)^2$ is the Koebe function, and the limit inferior is over $f\in S$ converging to $K$ locally uniformly on compact subsets. Concretely: (a) decide, in the pairs where it remains unknown — all pairs with $m<n$ except $(2,4)$ and $(3,4)$ (known numerically below the conjectured value) apart from the pairs with $m$ even and $n$ odd (where $\sigma_{mn}=B_{mn}=0$ is known), and the pairs with $m>n$, $m$ odd, $n$ even, with $n/m$ in the critical region above $0.8194$, together with the finitely many pairs $m<81$, $m$ odd, $n$ even, with $(m+1)/2<n\le0.8194\,m$ — whether $\sigma_{mn}$ equals $B_{mn}=\min_{\theta\in\mathbb{R}}\frac{n\sin\theta-\sin n\theta}{m\sin\theta-\sin m\theta}$, as conjectured by Bombieri (equation (5.1) of the source, p. 51); and (b) compute $\sigma_{mn}$ exactly for the pairs for which only the general bounds $0\le\sigma_{mn}\le B_{mn}$, or numerical approximations, are currently known (besides the single exact value $\sigma_{32}=(e-1)/4e$ and the trivial zeros). The two facets are the qualitative and the quantitative form of one objective: over a given pair, (b) determines the value and thereby settles (a), whereas (a) can be settled by exhibiting a counterexample family without determining the value, so their pair sets are not identical — (a) excludes the numerically decided pairs $(2,4)$ and $(3,4)$ that (b) still covers. The objective is to determine $\sigma_{mn}$ exactly and to decide whether $\sigma_{mn}=B_{mn}$ for the remaining undecided pairs.

References

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