PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 1 1 2, Number 3, July 199 1
LOWER ESTIMATE IN LITTLEWOOD'S CONJECTURE ON THE MEAN SPHERICAL DERIVATIVE OF A POLYNOMIAL AND ITERATION THEORY A.
E. EREMENKO
(Communicated by Clifford J. Earle, Jr.)
ABSTRACT. Lewis and Wu have proved J. E. Littlewood's conjecture of the
growth of the spherical derivative of polynomials. We apply the theory of
complex iteration to show that this conjecture is sharp in a qualitative sense.
Let f be a complex polynomial. Denoted by p ( f ) =
spherical derivative, let D be the unit disc, and set
if'/
1(1
+ lf21)
its
Littlewood [L] remarked that @(f) 5 nJii for all polynomials of degree
n . He asked for good upper and lower bounds for the quantity $(n) =
sup{@(f ) : deg f = n) and conjectured that
4
-
for some a < . It was proved in [ESl] that $(n) = o ( f i ) , n cx, and later
2-264
J. Lewis and J. M. Wu [LeW] obtained the estimate ( 1 ) with a = 2-' by improving the potential-theoretic method of [ESl, ES2]. Upper bounds for
$(n) have very interesting consequences in value distribution theory of entire
functions [L, ES 1, ESZ].
The lower estimate in [L] was $(n) 2 c(d)(log n)li2-* for every 6 > 0 .
Then W. Hayman [HI improved this to $(n) 2 c log n .
We shall prove in this note that $(n) 2 cnK for some K > 0 . To this end we
use complex iteration theory (for a brief account of this theory see for example
[B, EL]). Denote by f n the nth iterate of a polynomial f . The Julia set J(f )
is the closure of repelling periodic points of f . A polynomial f is called the
hyperbolic if f l ( z ) # 0 , z E J (f ) . In this case the closure of the orbit of
critical points of f does not intersect J (f ). For example f ( z ) = z2 + A is
Received by the editors December 1 , 1989 and, in revised form, June 30, 1990.
1980 Mathematics Subject Classification (1985 Revision). Primary 30C10; Secondary 30B20,
30D15.
@ 1991 Amerlcan Mathemat~calSociety
0002-9939191 $1.00 + $.25 per page
714
A. E. EREMENKO
hyperbolic if A is in the interior of the cardioid:
The orbit of the unique critical point of such a polynomial tends to the attracting
fixed point.
Let f be a hyperbolic polynomial. Consider a covering of J = J(f) by
closed squares %,of diameter r > 0 centered at the points zj E J and
satisfying int %,n int q.= 0 ,j # i . Denote by
the squares with the
centers zj having diameter 2r and sides parallel to those of gj.
Choose the
diameter r so small that 2Y1= U,
contains no critical values of iterates.
This is possible because critical values of fn are contained in the orbit of critical
points of f and f is hyperbolic. All branches of f-" are single valued and
univalent in
. They map
onto some regions Vk which cover J and
int Vkn int = 0 ,k # 1 . Choose ak in vk such that fn(ak)zj
=
for some
j . On applying the Koebe distortion theorem to each branch of f-" in
,
we obtain
9'
y1
y1
4L;'
(2)
area Vkx ~(f~)'(a,)l-~
and
(3)
I(~")'(~)I
~(f")'(~k)l, vk,
(x x y means clxI y 5 c2x with constants c, and c, depending only on
{ ). If z E V = UkVk then fn(z)E
and thus fn is bounded on V
uniformally in n . Futhermore we can suppose that J c 5.3= {z:lzl < 1) and
consequently V c 9 for large n . Using (2)and (3)we obtain
It is known that for every hyperbolic polynomial f and every t E R , the
following limit exists:
P(t)= P/(t)=
f log
C
l(fn)'(z)l-'.
This limit is called pressure and it does not depend on a. (See [Bo, R]). It
follows from (4)that
@(fn)2 cexp(nPf(l))
= c(degfnlK,
where ic = Pf(l)/
log deg f . D. Ruelle [R] proved that t H Pf(t)is a decreasing
convex function. Moreover Pf(6)= 0 ,where 6 is the Hausdorff dimension of
J . The derivative dPf/dtis negative at t = 6 . We conclude that Pf(l)> 0
if 6 > 1 . This is the case for example if f ( z ) = z2+ A and A is in the interior
of the above-mentioned cardioid C .
LOWER ESTIMATE IN LITTLEWOOD'S CONJECTURE
715
Note that a slightly more accurate computation shows that @(f") x (deg fn ) K
where K = Pf(l)/log deg f.
Two natural questions arise:
(1) What is sup{P,(l): f ( z ) = z2 + 1) ?
(2) Define the Littlewood order of a sequence of polynomials P,, by
lim log @(P,)/ log deg Pn.
n+m
Is it true that the largest possible order occurs if P, are iterates of a
polynomial f?
P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc.
11 (1984), 85-141.
R.
Bowen, Equilbrium states and the ergodic theory of Anosov diffeomorphisms, Lecture
[Bo]
Notes in Math., vol. 470, Springer-Verlag, Berlin and New York, 1975.
[EL] A. Eremenko and M. Lyubich, Dynamics ofanalytical transformations, Algebra and Analysis
1 (1989), 1-70 (Russian); English transl.: Leningrad Math. J. 1 (1990), 563-634.
[ESl] A. Eremenko and M. Sodin, On a conjecture of Littlewood and value distribution of entire
functions, Funct. Anal. Appl. 20 (1986), 71-72.
Proof of a conditional theorem of Littlewood on the distribution of values of entire
[ES2] ,
functions, Math. USSR Izvestiya 30 (1988), 395-402.
W. Hayman, On a conjecture ofLittlewood, J. Anal. Math. 36 (1979), 75-95.
[HI
J. E. Littlewood, On some conjectural inequalities with applications to the theory of intergral
[L] functions, J. London Math. Soc. 27 (1952), 387-393.
[LeW] J. L. Lewis and Jang-Mei Wu, On conjectures ofArakelyan and Littlewood, J. Anal. Math.
50 (1988), 259-283.
[R] D. Ruelle, Repellers for real analytic maps, Ergodic Theory Dynamical Systems 2 (1982),
99-107.
[B]