On Littlewood’s constants

D. Beliaev^{∗}  S. Smirnov^{†}^{‡}

Abstract

In 1952 Littlewood studied the following problem: what is the best constant α such that for any polynomial f of degree n the areal integral of its spherical derivative is at most const · n^{α}?

We show that α is equal to another constant β independently studied by Littlewood, namely the growth rate of the length of Green’s lines. This greatly improves known estimates on α.

1 Introduction

In this paper we study the growth rate as n → ∞ of the quantity

A_{n} = sup ∫_{𝔻} \frac{|g′|}{1 + |g|^{2}} dm,

where supremum is taken over all polynomials g of degree n, 𝔻 is a unit disc {|z| < 1}, and m denotes two-dimensional Lebesgue measure. We are interested in the best α such that A_{n} ≲ n^{α} (which means that for every ε > 0 there is a constant C_{ε} with A_{n} ≤ C_{ε}n^{α+ε}). In [L1] Littlewood observed that 0 ≤ α ≤ 1/2 and conjectured that α < 1/2.

It is quite easy to show that for any rational function g of degree n

∫_{𝔻} \frac{|g′|}{1 + |g|^{2}} dm ≤ const\sqrt{n}.

Note that integrand is a modulus of spherical derivative g′_{σ} of g (i.e. derivative with respect to spherical metric) and that in 𝔻 spherical measure dm_{σ} is comparable to Lebesgue measure dm. So our integral can be estimated by

∫_{𝔻} |g′_{σ}|dm_{σ} ≤ ∫_{ℂ} |g′_{σ}|dm_{σ} ≤ (∫_{ℂ} |g′_{σ}|^{2}dm_{σ})^{1/2} (∫_{ℂ} dm_{σ})^{1/2}
= (2πn)^{1/2}(2π)^{1/2} = 2π\sqrt{n},

∗Department of Mathematics, Royal Institute of Technology (KTH), 100 44 Stockholm, Sweden. beliaev@math.kth.se

†Department of Mathematics, Royal Institute of Technology (KTH), 100 44 Stockholm, Sweden. stas@math.kth.se

‡The second author is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation.

here we use that rational function of degree n maps complex sphere to itself n-to-1 so the area of the image is n times area of the sphere. In particular this argument shows that α ≤ 1/2.

Littlewood’s conjecture was proved in [LW] by Lewis and Wu, who improving upon the work [ES] of Eremenko and Sodin obtained an explicit upper estimate α < 1/2−2^{−264}. Later Eremenko in [E] obtained a positive lower bound on α.

Following works by Eremenko and Sodin [ES] and Lewis and Wu [LW] we exploit connection between this problem and extremal behavior of harmonic measure.

Our main result is that α is related to the growth rate of Green’s lines length. In the case of simply connected domains Ω we define β_{Ω} as

lim sup_{ε→0} \frac{log length\{z : G(z) = ε\}}{log 1/ε} ,

where G is the Green’s function with pole at infinity and we define

β = sup β_{Ω},

where supremum is taken over all simply connected domains Ω. In the non simply connected case one needs more elaborate definition and we use the multifractal analysis technique.

For a given domain Ω we define the packing spectrum π_{Ω}(t) as

sup {q : ∀δ > 0 ∃ δ − packing {B} with ∑ diam(B)^{t}ω(B)^{q} ≥ 1} ,

where ω is the harmonic measure in Ω. Note that this definition is valid for any domain with compact boundary and for t = 1 it is analogue of β_{Ω} (see [M] for the proof of π_{Ω}(1) = π_{Ω}(1) for simply connected Ω) .

We define the universal spectrum π(t) as the supremum of π_{Ω}(t) over all planar domains Ω with compact boundary.

The main result of this paper is the following theorem:

Main Theorem. For any positive ε there exists a constant c = c(ε) such that

A_{n} ≤ cn^{π(1)+ε}.

Equivalently

α ≤ π(1).

Remark. Our proof actually implies that for t ∈ [0, 2]

∫_{𝔻} ( \frac{|g′|}{1 + |g|^{2}} )^{t} ≤ cn^{π(2−t)+ε}.

Of interest to us are also π_{p}(t) and π_{p,sc}(t) which are suprema of π_{Ω}(t) over all domains of attraction to infinity for polynomial mappings and simply connected domains of attraction to infinity for polynomial mappings correspondingly (see [CG] for background on complex dynamics). It is clear that π_{p,sc} ≤ π_{p} ≤ π, but a priory they might differ.

In [E] Eremenko essentially proved that π_{p,sc}(1) ≤ α (he works under assumption that polynomials are hyperbolic, but it can be easily avoided). Binder, Makarov, and Smirnov in [BMS] showed that π_{p,sc}(t) = π_{p}(t) for t > 0. Recently Binder and Jones [BJ] announced the proof of the identity π_{p} = π, which together with our theorem completes the circle:

α ≤^{MainThm.} π(1) =^{[BJ]} π_{p}(1) =^{[BMS]} π_{p,sc}(1) ≤^{[E]} α.

There is yet another growth rate that is related to α and was also studied by Littlewood. The growth rate γ of coefficients of univalent functions in 𝔻^{−} is defined

sup_{ϕ} lim sup_{n→∞} \frac{log |b_{n}|}{log n} + 1,

where the first supremum is taken over all functions ϕ(z) = z+∑_{n=1}^{∞} b_{n}z^{−n} that are univalent in 𝔻^{−}. Littlewood proved [L2, LP] that β ≥ γ. Much later Carleson and Jones [CJ] showed that γ = β. Summing it all up, we arrive at:

Corollary:

α = β = γ = π(1) = π_{p,sc}(1).  (1)

Note that a well-known conjecture (see [CJ, M, K, P1]) states that π(t) = (2 − t)^{2}/4 for |t| ≤ 2, and particularly α = β = γ = 1/4. The best known estimates for β are

0.17 <^{[P]} β <^{[GP]} 0.4884,

so the same estimates hold for α, which is a significant improvement over previously known 1.11 · 10^{−5} ≤ α ≤ 1/2 − 2^{−264} (see [BS] and [LW]).

Connection to the value distribution of entire functions Before giving the proof we would like to remark that the reason for

Littlewood’s interest in this problem was the following striking corollary of his conjecture (see [L1]):

Littlewood’s conditional theorem Assume that α < 1/2. If f is an entire function of order 0 < ρ < ∞ then for any 0 < θ < 1/2 − α there is a “small” set S such that “almost all” roots of any equation f(z) = w lie in S. Namely

\frac{Area(S ∩ B(R))}{Area(B(R))} = O ( \frac{1}{R^{2θρ}} ) , R → ∞,

and for all w

\frac{#\{z ∈ B(R) \ S : f(z) = w\}}{#\{z ∈ B(R) : f(z) = w\}} = O ( \frac{1}{R^{ρ(1/2−α−θ)}} ) , R → ∞,

where B(R) is a disc of radius R centered at the origin.

Acknowledgment

We would like to thank M. Sodin for useful discussions and comments on the preliminary version.

2 Proof of The Main Theorem

It is a standard fact (for details see [M] and [F]) that π(t) is finite, convex, and strictly decreasing on [0, 2], and hence for any small δ we can choose ε so small that

π(1 − 2ε) − π(1) < δ.

We will also use a more elaborate fact (which follows from multifractal formalism and fractal approximation – see [M]), that there is a constant const(t, ε), such that for any disjoint collection of cubes {Q} of size ≤ 1 one has

∑_{Q} ω(Q)^{π(t)}l(Q)^{t+ε} < const(t, ε).

Let g be a polynomial of degree at most n and a_{i} its zeros. Consider a set where |g| is big, i.e. |g| ≥ n. We can easily estimate integral over this set by

\begin{matrix}∫_{𝔻∩\{|g|≥n\}} \frac{|g′|}{1 + |g|^{2}} ≤ n^{−1} ∫_{𝔻} \frac{|g′|}{|g|} = n^{−1} ∫_{𝔻} |(log g)′| \\ = n^{−1} ∫_{𝔻} |∑_{i=1}^{n} \frac{1}{z − a_{i}} | ≤ n^{−1} · 2πn = 2π. \end{matrix}  (2)

Now consider a complementary set where |g| is small, which is contained inside the disc of radius 3/2

Ω := \{z : |g(z)| < n, |z| < \frac{3}{2} \} ,

and let W = {Q_{j}} be a Whitney decomposition of Ω. We note that

dμ(z) = 4n^{−1} \frac{|g′(z)|^{2}}{(1 + |g(z)|^{2})^{2}} dxdy,

is the Riesz measure associated with the nonnegative subharmonic function

u = \frac{log(1 + |g|^{2})}{n} .

Then by Riesz representation formula

μ(B(z, r)) ≤ \frac{c}{2π} ∫_{0}^{2π}(u(z + 2re^{iθ}) − u(z))dθ.  (3)

Hence for every cube Q_{j} we have

μ(Q_{j}) ≤ c \frac{log(1 + n)}{n} .

Fix a cube Q_{j} such that Q_{j} ∩ 𝔻 ≠ ∅ and denote by ξ_{j} a point at ∂Ω such that d(ξ_{j}, Q_{j}) ≤ 2l(Q_{j}). Then from (3)

\begin{aligned}μ(Q_{j}) & ≤ & \frac{c}{2π} ∫_{0}^{2π}(u(ξ_{j} + 8l(Q_{j})e^{iθ}) − u(ξ_{j}))dθ \\ & ≤ & max_{z∈B(ξ_{j} ,8l(Q_{j}))} c[u(z) − u(ξ_{j})]^{+} . \end{aligned}  (4)

Denote by G(z) the Green’s function for ℂ\\overline{Ω} with pole at infinity. Extend G to a continuous subharmonic function in ℂ by setting G = 0 on Ω.

By the maximum principle for domain ℂ\\overline{Ω} we obtain G(z) ≥ log|2z/3| for any z ∈ ℂ \ Ω, hence

G(z) ≥ log \frac{4}{3} for |z| = 2.

By the maximum principle for domain 2𝔻 \ \overline{Ω}, we have

u(z) − u(ξ_{j}) ≤ M_{2}G(z) (log \frac{4}{3} )^{−1} for |z| ≤ 2,

where M_{2} = max_{|z|=2} u(z).

If we let z_{j} be a center of Q_{j} and ω be a harmonic measure on ℂ \ \overline{Ω} with pole at infinity then by previous inequality and (4) we have

μ(Q_{j}) ≤ max_{z∈B(ξ_{j} ,8l(Q_{j}))} c[u(z) − u(ξ_{j})]^{+}

≤ c (log \frac{4}{3} )^{−1} M_{2} max_{z∈B(ξ_{j} ,8l(Q_{j}))} G(z) ≤ cM_{2} max_{z∈B(z_{j} ,16l(Q_{j}))} G(z).

By Harnack’s inequality right hand side is less than

cM_{2}3 ∫_{∂B(z_{j} ,32l(Q_{j}))} G(z) \frac{|dz|}{2π32l(Q_{j})} ,

which by Riesz representation formula equals to

cM_{2} ∫_{0}^{32l(Q_{j})}\frac{ω(B(z_{j}, t))}{t} dt ≤ cM_{2}ω(B(z_{j}, 32l(Q_{j}))).

So finally we have

μ(Q_{j}) ≤ const M_{2} ω(B(z_{j}, 32l(Q_{j}))).  (5)

By Schwarz’s inequality we have

∫_{Ω} \frac{|g′|}{1 + |g|^{2}} dxdy = n^{1/2} ∑_{Q_{j} ∈W} n^{−1/2} ∫_{Q_{j}} \frac{|g′|}{1 + |g|^{2}} dxdy

≤ \frac{1}{2} n^{1/2} ∑_{Q_{j} ∈W} (∫_{Q_{j}} \frac{4|g′|^{2}}{n(1 + |g|^{2})^{2}} dxdy)^{1/2} (∫_{Q_{j}} dxdy)^{1/2}

≤ \frac{1}{2} n^{1/2} ∑_{Q_{j} ∈W} μ(Q_{j})^{1/2}l(Q_{j}) = \frac{1}{2} n^{1/2} ∑_{Q_{j} ∈W} μ(Q_{j})^{1/2−π(1−2ε)}μ(Q_{j})^{π(1−2ε)}l(Q_{j})

≤ Cn^{1/2} ( \frac{log(1 + n)}{n} )^{1/2−π(1−2ε)} ∑_{Q_{j} ∈W} μ(Q_{j})^{π(1−2ε)}l(Q_{j})

≤ Cn^{π(1)+δ} ∑_{Q_{j} ∈D} μ(Q_{j})^{π(1−2ε)}l(Q_{j}),

where D is the family of all dyadic squares with the side length less then 32 that intersect Ω, ε is a small positive number, and C is a constant. By (5) we can estimate the last sum

Cn^{π(1)+δ} ∑_{Q_{j} ∈D} μ(Q_{j})^{π(1−2ε)}l(Q_{j})

≤ C M_{2}n^{π(1)+δ} ∑_{k=1}^{∞} ∑_{l(Q_{j})=1/2^{k}} ω(Q_{j})^{π(1−2ε)}l(Q_{j})^{(1−2ε)+ε}l(Q_{j})^{ε}

= C M_{2}n^{π(1)+δ} ∑_{k=1}^{∞} \frac{1}{2^{kε}} ∑_{l(Q_{j})=1/2^{k}} ω(Q_{j})^{π(1−2ε)}l(Q_{j})^{(1−2ε)+ε}

≤ C M_{2}n^{π(1)+δ} ∑_{k=1}^{∞} \frac{1}{2^{kε}} const(ε) ≤ n^{π(1)+δ}const(ε)M_{2}.

(6)

Now assume the following dichotomy: For any ε > 0 there exists a constant const(ε) such that for any polynomial g of degree n

M_{2} ≤ const(ε)n^{ε},  (7)

or

|Ω| ≤ 1/n.  (8)

Then if (7) holds then the desired estimate follows from (6):

∫_{Ω} \frac{|g′|}{1 + |g|^{2}} ≤ constn^{π(1)+ε+δ}.

But both ε and δ can be made arbitrary small we have the desired estimate.

If (8) holds then even better estimate follows from Schwartz’ inequality:

∫_{Ω} \frac{|g′|}{1 + |g|^{2}} ≤ (∫_{Ω} \frac{|g′|^{2}}{(1 + |g|^{2})^{2}} )^{1/2} (∫_{Ω} 1)^{1/2}

≤ (∫_{ℂ} |g′_{σ}|)^{1/2} \sqrt{|Ω|} ≤ \sqrt{2πn|Ω|} ≤ \sqrt{2π}.

Therefore it remains to prove the dichotomy.

Proof of the dichotomy. Assume that M_{2} > n^{ε}. Recalling the definition M_{2} = sup_{|z|=2} log(1 + |g|^{2})/n, we deduce that sup_{|z|=2} |g| > exp(n^{1+ε}) so the set Ω where |g| < n cannot have big measure.

We can write g as g = P Q, where

P (z) = λ ∏_{|a_{i}|>4}(z − a_{i}), Q(z) = λ ∏_{|a_{i}|≤4}(z − a_{i}).

Let m be the degree of P, then

log (|λ| ∏_{|a_{i}|>4} |a_{i}|)−m log 2 ≤ log |P (z)| ≤ log (|λ| ∏_{|a_{i}|>4} |a_{i}|)+m log 2,

when |z| ≤ 2. Since |Q(z)| < 6^{n} for |z| ≤ 2, it follows that

inf_{z∈Ω} log |P (z)| ≥ sup_{|z|=2} log |P (z)|3^{−n} ≥ log (exp(n^{1+ε})3^{−n}6^{−n}) ≥ \frac{1}{2} n^{1+ε},

if n is sufficiently large. Since log |P Q| = log |g| < n in Ω we can write

log |Q(z)| ≤ log |P Q| − log |P | ≤ n − \frac{1}{2} n^{1+ε} ≤ − \frac{1}{4} n^{1+ε}, z ∈ Ω

if n is large enough. Therefore, Ω is contained in the union of disks {z : |z − a_{i}| ≤ exp(−n^{ε}/4)}. Hence

|Ω| ≤ nπ exp(− \frac{1}{2} n^{ε}) ≤ 1/n when n is sufficiently large.

This proves the dichotomy for n > N(ε), for degree n bounded from above by N(ε) the dichotomy is easy by compactness argument (and anyway, it suffices to prove the estimate for polynomials of sufficiently large degree). This completes the proof of the main theorem. □

References

[BS] I.N. Baker, G.M. Stallard, Error estimates in a calculation of Ruelle. Complex Variables Theory Appl. 29 (1996), no 2, 141-159.

[BJ] I. Binder, P. Jones, private communication.

[BMS] I. Binder, N. Makarov, S. Smirnov, Harmonic measure and polynomial Julia sets, Duke Math. J. (to appear).

[CG] L. Carleson, T. Gamelin, Complex dynamics. Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993.

[CJ] L. Carleson, P. Jones, On coefficient problems for univalent functions and conformal dimension. Duke Math. J. 66 (1992), no. 2, 169–206.

[E] A. Eremenko, Lower estimate in Littlewood’s conjecture on the mean spherical derivative of a polynomial and iteration theory. Proc. Amer. Math. Soc. 112 (1991), no. 3, 713–715.

[ES] A. Eremenko and M. Sodin, A proof of the conditional Littlewood theorem on the distribution of the values of entire functions. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 2, 421–428, 448; translation in Math. USSR-Izv. 30 (1988), no. 2, 395–402.

[F] K. Falconer, Techniques in fractal geometry. John Wiley & Sons, Ltd., Chichester, 1997.

[GP] A.Z. Grinshpan, Ch. Pommerenke, The Grunsky norm and some coefficient estimates for bounded functions. Bull. London Math. Soc. 29 (1997), no. 6, 705–712.

[K] P. Kraetzer, Experimental bounds for the universal integral means spectrum of conformal maps. Complex Variables Theory Appl. 31 (1996), no. 4, 305–309.

[L1] J. Littlewood, On some conjectural inequalities, with applications to the theory of integral functions. J. London Math. Soc. 27, (1952). 387–393.

[L2] J. Littlewood, On the coefficients of schlicht functions. Quart. J. Math. Oxford Ser. 9 (1938), 14–20.

[LP] J. Littlewood, R. Paley, A proof that an odd schlicht function has bounded coefficients, J. London Math. Soc. 7 (1932), 167–169.

[LW] J. Lewis and J.M. Wu, On conjectures of Arakelyan and Littlewood. J. Analyse Math. 50 (1988), 259–283.

[M] N. Makarov, Fine structure of harmonic measure. St.Petersburg Math. J. Vol. 10 (1999), No. 2, 217–268.

[P1] Ch. Pommerenke, The integral means spectrum of univalent functions. (English. Russian summary) Zap. Nauchn. Sem. S.- Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 237 (1997), Anal. Teor. Chisel i Teor. Funkts. 14, 119–128, 229; translation in J. Math. Sci. (New York) 95 (1999), no. 3, 2249–2255.

[P2] Ch. Pommerenke, Boundary behaviour of conformal maps. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 299. Springer-Verlag, Berlin, 1992.