Photocopying permitted by license only licensebyGordon and Breach Science Publishers Printed inMalaysia
Small Values of Polynomials"
Cartan, P61ya and Others
D.S. LUBINSKY
Department
ofMathematics, WitwatersrandUniversity, Wits2050, SouthAfricaE-mail:[email protected] (Received20June1996)
Let P (z)beamonicpolynomialofdegreen, andc, e >0.Aclassic lemma ofCartanasserts that the lemniscateE(P; e) := {z IP(z)l<en}canbe coveredbyballsBj, <j <n, whose diametersd(Bj)satisfy
p
(d(Bj))
<e(4s).
j=l
Forot 2, this shows thatE(p; e)hasan areaat mostre(2e)2.P61yashowed inthiscase that thesharpestimate isrre2.Wediscusssomeoftheramificationsof these estimates, as wellas someof their close cousins, forexamplewhenPisnormalizedtohaveLpnorm onsome circle, andRemez’ inequality.
Keywords: Polynomials; small values;Cartan’slemma;P61ya; Remez; capacity.
1991MathematicsSubjectClassification: Primary 30C10, 41A17;Secondary31A15, 30C85, 41A44.
1
INTRODUCTION
Onhowlargeasetcan apolynomialbesmall? Thissimple questionand its cousins has fascinatedmathematicians of the status of
H. Cartan, G. P61ya
andP. Erd6s;its ramificationsrangefrom thetheoryof entire functions and potential theorytorationalapproximation andorthogonal polynomials.In
thispaperwe shalldiscuss,..someof these results.The firststepis normalization of thepolynomial.The obvious choices are normalizing
P
tobe monic, that is, to haveleadingcoefficient 1:P(z) z" +... (.)
199
or tohave some norm1,forexample,for some fixedr > 0,0 < p <
[Ie[lLp(Izl-r)
1.(1.2)
Let
usbeginwith theformer,namelymonicpolynomials.The classic1928 resultofCartan [4],introduced with a view tostudyingminimummoduliof holomorphicfunctions, stillreigns supreme. LetP
be monic ofdegreen, and e,ot > 0. The lemma asserts thatp
E(P;
(z "lP(z l
_<(1.3)
j=l
where p < n,and B1, B2,...
Bp
areballs with diametersd(Bj)
satisfyingE
p(d(Bj))
<e(4e)a" (1.4)
j=l
What isremarkable is that this estimate doesnotdependon thedegreen of
P.
The naiveapproachofplacingballs of radius e centred on each of the zerosofPandusingthese to coverE
(P;)
givesthe much weaker estimate ofn(2e)
In
particular, whenot 2, andmeas2 denotesplanar Lebesguemeasure, (1.4)becomesmeas2(E(P; e))
<7re(2e) 2.
In
thisspecialcase, a result ofP61yafrom 1928assertsthat there isthe sharp estimatemeas2(E(P; s))
<_ rs2withequalityiffP (z)
(z a) n,
somea 6C.
It
would be a realchallengeto determine forgeneralor, thesharpconstant that shouldreplacee4 in(1.4). For
ot 1,theconjecturedsharpconstant is4.Likeallgreatinequalities,Cartan’shasinspiredextensions.
In
Section2, weshall state andproveanextensionwhered(Bj)
isreplaced byh(d(Bj))
for suitableincreasingfunctions h. This form is useful instudyingthin sets that arise inpotentialtheoryandPad6approximation.
In
Section3, we shallprove P61ya’s inequality.Because
it’sproof
is so closely linkedtoGreen’s theorem,and the latter is tied toplanarmeasure, P61ya’s methods do not seem to havemany generalizations. Orhave they been missed?The alternative normalization
(1.2)
is oftenmoreuseful than the monic one.For
example,inconvergence theoryof Pad6approximation,oneiscalled on to estimate the ratioIIP [IL(Izl--r) /[ e (z)
outsideassmall a set as possible.Ifoneadoptsthe normalization
(1.2)
with p cx),then at least the numerator is taken careof, and then one wants to seehowlargecanbe theseton whichP (z)
issmall. Untilquite recently, thishasalwaysbeen via capacitary estimates orCartan’s
lemma: One splits, following Nuttallandothers,--: cR(z)S(z).
Izjl<2r Izjl>2r Zj Since for
zl
r, zj > 2r,andfor
Izl
r,Izj
2r,we seethat
1 3
Iz
zj < 3rIIPIILo(Izl=r)/IP(z)l <_ (3 max{l, rI)/IR(z)l.
As R
is monic, one canapply Cartan’slemma todeduce thatIIPIIro(Izl--r)/le(z)l (3 max{l, r}/e)
nfor ]zl < r outside a setthat can be covered by balls
{Bj}
admitting the estimate(1.4).
Thisprocedure,ofseparatingzeros intosmall andlargeones, is useful in manycontexts, and has been usedbymathematiciansof the status ofNuttall, Pommerenke,
Goncar,
Stahl and others but does not admiteasyextension to several variables. Motivatedby the latter,A. Cuyt, K.
Driver and the author[5]
were led to directlyconsider the normalization(1.2).
Thisand the Walsh-Bernsteininequality, whichboundspolynomials anywherein the planein terms of their maximummodulus on somegivensetthatisnot"too
thin",very quickly givefor the restricted lemniscateE(P;r;
) := {z "lzl
<_ r,IP(z)I
<_the estimates
cap(E(P;
r;e))
<_ 2re(1.5)
and
meas2(E(P;
r;e))
<7r(2re)
2(1.6)
andthat these aresharpfor each r,n.
Here
cap islogarithmic capacity(we
shall define that in Section2).
We shall discuss this simple approach in Section3, givingalso some of itsLp
extensions.In
Section4,we review some of theimplicationsofRemez’
inequalityfor smallvalues ofpolynomials.In
Section5,weshallbrieflyreviewsome of the multivariate extensionsofCartan’slemma.Here
it is difficulttodecide what is a monicpolynomial!Moreover,
the measures of thinnessof sets become quite complicated, and the fact that the lemniscate is unbounded leads to difficulties.2
THE CARTAN APPROACH
We
beginby presenting Cartan’sclassical argumentin ageneral form,the powerof which lies in thearbitraryvaluesassignedtothe numbersrj.TI-mOREM2.1 Let 0 <
ra
< r2 < <rn
and P(z)beamonicpolynomialof
degree n. Thereexistpositive integers p < n{,j}P
j=l andclosed ballsBj }P.
j--1 such that.1
-]- )2"-- +
)p n(ii)
d(Bj) 4rzj,
1 < j < p.(iii)
z,
Ie(z l <_
rjc
j=l j=l
(2.1)
Proof We
divide this into foursteps:STEP
1"We
show that there exists )vl < n and a circleC1
of radius rzl containing exactly)Vl zeros ofP,
counting multiplicity.For
supposesuchacircle does not exist. Then anycircle Cofradiusrl containing1 zero ofP
contains at least 2. Theconcentric circleofradiusr2 contains2, so must contain3 (otherwisewe couldchoose )l 2 andC1
tobe thiscircle). Continuing in thisway, weeventuallyfindthat the circle concentric with C and radius
rn
mustcontain n+
1 zeros ofP, which isimpossible.
SxEv
2:We
rank the zeros ofP.Choose the largest ,1 with the property in
Step
1, and letC1
be the corresponding circle. Call the.1
zeros of P insideC1
zeros of rank )l.Next,
applyingtheargumentofStep
1 totheremainingnX2
zeros ofP,
weobtain alargest positive integer )2 < )1 and a circle
C2
containing exactly2
of the zeros ofPoutsideC1.
Call those zeros insideC2
zeros of rank)2.Continuingin thisway,we find p <nlargest integers)l > )2 _> >
)p
andcorrespondingcirclesCj
of radiusrzj containingexactly)j
zeros ofP outsideC1
UC2
tO...tOCj-1. Moreover,
asweeventuallyexhaust the zeros,.1 +
,2-+-"""-at-)p
n.STEP
3" We provethat ifSis a circleof radiusrz
containing at least)zeros ofP,thenatleast one of these zeros has rank at least.
First if
S
contains more than)1zeros, then at leastonemustlieinC1
andsohave rank )l > ).
(If
not,wewould obtain a contradictiontothe choice of)1 being aslargeaspossible).Next
supposethat)j
> ) >;j+l,
some j. Ifany of the zeros insideSlies in C1,C2,...,Cj
then these have rank>
)j
> ;k, asrequired.If all the zeros lie outside these former j circles, then theprocessofStep
1yieldsacircle with> ) >,j
+ zeroscontradictingthe choiceofj+l beingaslargeaspossible.STEP
4: CompletetheproofLet
Bj
bethe(closed)ballconcentricwithCj
but twice the radius, so thatd(Bj) 4rzj,
1 < j < p. Fixz
6C\ I,-J;_ nj. We
claimthat a circleS,
centrez, radius rz, can contain at most) 1 zerosofP. Forifitcontained atleast
,
thenbyStep
3,atleast one,u say,wouldhave, say,rank)j
> ), andsolie inCj
and alsoiiathe concentric ballBj
of twice the radius. Then thefact thatzBj
andulies insideCj
forcesIz ul
>dist(C\Bj, Cj) rz
>rz
contradictingourhypothesisthat
S
containsu.Finally rearrange the zeros in order of increasing distance from
z
asz
l,z2 Zn.Now
the circle centre z, radiusrj can contain atmost j 1 zeros ofP,
andthese couldonlybeZl,z2 zj- soIz-zjl
> rj.Thus
IP(z)l--]II(z
j=lz) >Hrj.
j=lnI
wonder if anon-geometricproofwill everbe found as an alternative to Cartan’sbeautiful oneabove.Now
by choosing{rj
in variousways,wecan obtainall sorts of estimates.COROLLARY
2.2 ThenLet
P be amonicpolynomialof
degree n and e,c > O.where
p
(d(Bj))
<_e(4e) a.
j=l
Proof We
chooseThen
rj Ej1/a
(n !)-
1 1 < j <nand with
.{.j
as inTheorem2.1,p p p
(d(Bj))
a 4a(rzJ )a (4e)a(n!)-l/n Z
jj--1 j=l j=l
(4e)a
<e(4e)a
bytheelementary inequalityn >
(n /e)
n []For some applications, one needs to replace
d(Bj)
by h(d(Bj))
forsomepositivemonotoneincreasingfunction
h(t)
that has limit 0atO. Such functionsarecloselyassociated withHausdorffh content ormeasure.Let h [0,c)
--+ [0,cxz)
be a monotoneincreasing non-negativefunction with limit0atO.Thecorresponding Hausdorffcontent,is defined forE CC,
byh
c(E)
infh(d(Bj))’ECUBj
j=l j=l
Note that only balls
Bj
are considered for covering E. The Hausdorff h-measure is definedsimilarly: when taking the inf, one restricts eachBj
to haved(Bj)
< and then lets 3 --+ 0/.For
ourpurposes,we note only that Hausdorff h-measure and h-content vanish on the same sets. The classic reference isRogers [19]. In
thislanguage, if we leth,(t) ,
the estimate of Corollary2.2maybe written asha -c(E(P; e))
<e(4e) .
In
formulatingour result forh c,we need thegeneralizedinverse of a continuous function g [0,1] --+N,
definedbyg[-1](s)
min{t"
g(t)s},
s eg([0, 1]).
THEOREM2.3 Let h [0,1] --+ [0,
cxz)
be strictly increasingin [0, 1]with limit0atO,andabsolutelycontinuous ineachclosed subintervalof
(0, 1].(I) Assume
h(t)dt < cxz.
(2.2)
Let
( fo )
g(t) exp
-h---
]logulh’(u)du
e (0,1).
Then
for
n > 1 andmonicpolynomials Pof
degreen,h-
c(e(e;
_<h(g[-11(43)),
3 e 0,---
In particular, theboundsdecaysto0as --+ 0+, since
(2.3)
(2.4)
lim
g[-1](t)
0.(2.5)
t-+0+
(II)
Conversely,if
there existsa monotoneincreasingfunction
X [0, 1] --+[0,
cxz)
with limitO.at
O, such thatfor
n > 1,3 (0,1)
andmonicpolynomials P
of
degreen,h-
c(E(P; 3))
<X(3),
3 e (0,1) (2.6)
while
for
some0 <fl
< 2,h(t)/t
ismonotonedecreasingnearO,then
(2.2)
holds.Under additional conditions, we canreplacetheabove implicitestimate by something simpler:
COROLLARY 2.4
Suppose
thereexistsA
> 1 such thath(U)du < A_l
0,-
(2.7)h(t)llogtl u Then(2.4)may bereplaced by
h
c(E(P; 3))
<h((43)l/a),
3 [0,2A-2].
(2.8)COROLLARY 2.5
Ifh(t) (log )-, ’
> 1, then(2.4)
may bereplaced by hc(E(P 3))<(Y.-1) -_ h(43),
3 6[ 0,-7 41) (2.9)
It is noteworthy that only the behaviour of h in an arbitrarily small neighbourhoodof 0 isimportantin applications, for which 3 above isusually close to 0. Thus onemay always modify h awayfrom 0 to ensure its definition throughout (0,1).
WeturntotheProof of
Theorem 2.3(I) Let
h[-1]be the(ordinary)inverse ofh,defined atleast on[0, h(1)]
byh(h[-ll(u))
u.FixH
6 [0, 1]and setrj
lh[-1] (
jn l<j<n._Then if
{)j
areas in Theorem2.1,P P P
h(d(Bj)) Z h(4rzj)= h(H) )j
h(H).j=l j=l j=l n
(2.10)
Moreover,
H
rj 4-nexp log h[-11J
h 4-nj--1 j=l
(H)
> exp(I) where, bythemonotonicityofh, h[-1],
f0" ( )
I "= logh[-1]
h(H)
du.The substitution
h(v) U--h(H)
givesnfo
h(H)
(logv)h’ (v)
dv nlog g(H).In summary,
H
rj > g(H)j=l 4
(2.11)
Thisworksonlyif theintegral definingg ismeaningful,whichwe now show:
An
integrationby parts givesl 0t
log g(t)
h(t)
(logv)h’ (v)
dv 1fo h(V)dv.
log h(t) v
(2.12)
Here
wehave used the fact thath(t)log has limit 0 at 0: this is an easy consequenceof theconvergenceof theintegralin(2.2).
Sowe see thatg is well defined. Theorem 2.1 and(2.11)
giveh_c
({z. ip(z)l
<(g
n < h(H).Now
g is continuous,h >_0,so we see from(2.3)
that0 < g(t)<_
t,#
0 so[
g(1)]
there exists(2.5)
holds and g([0, 1]) ___ [0, g(1)].Thusforeach 0,--T-
H
6 [0, 1]such that3 g(H)/4.Moreover,wemaychooseHg[-11(43).
Proof of
Corollary2.4 From (2.12),we seethat thebound given in(2.7)
is equivalent to-logg(t) <_ Allogtl, 0,
=,
g(t) > A= g[-ll(u)
_<u/A,
u [0,2-A].
[]Proof of
Corollary2.5A
calculationshows that g(t)
?’/(?’-1)andhenceg[-1](u)
u(?’-1)/’.
Onethen uses thespecificform of h. []In
the proof of Theorem2.3(1I)
and also in the next section, we shall need the notion of(logarithmic) capacity.There are at least fourequivalent definitions, but thesimplestisthefollowing: For compact E CC,
cap(E) lim min
IIPIIL(E>
n--cx) deg(P)=n, Pmonic
For
non-compactF
CC,
cap(F)
sup{cap(E)"
E CF,
Ecompact}.
For
aproof
that the above limit exists, and an introductiontocap,perhaps the best source is Chapter 16 of Hille [9].Deeper
treatments may be found in Carleson [3],Hayman
andKennedy [8] and Landkof[11]. What is particularly relevant for the purposes of this paper is that for monic polynomials P ande > 0,cap(E(P; s))
s.(The
inequalitycap(E(P; s))
<_ s iseasily provedfrom the definition of cap;the converse isalittlemore difficult,requiringthemaximummodulus principle).This identity shows that capisoften the naturalset functionto measuresmall values ofpolynomials.We
turn totheProof of
Theorem2.3(11) Assume (2.6)
holds.We
shall assume theintegral in(2.2)
diverges and derive a contradiction.We
use two results of S.J.Taylor [23]:
Because
ofthe assumedregularitycondition onh, thereexists acompactsetEof finitepositiveHausdorffh-measure,and hence also with 0 < hc(E)
< cxz.In
addition, E has so-calledpositivelowerspherical densityateach of itspoints.If theintegralin(2.2)
diverges,another theorem ofTaylorensuresthatcap(E) 0.But
thenfrom the definition of cap, for arbitrarilysmall e > O,andcorrespondinglysuitablen andPofdegreen,and hence
(2.6)
givesE
{z’le<zl
h-c<E h-c({z" le<zl })
SinceX has limit 0at0,wededuce thath
c(E)
0, acontradiction.So
theintegralin
(2.2)
mustconverge. []Theorem 2.3 and its corollaries are neater formulations of (presumably
new)
results in [14,15]. One
of their consequencesare explicit estimates relating cap and h c. Since any compact set E canbe contained in a lemniscate{z le<z)l
<_(cap(E) + 8)
n forarbitrarilysmall8,itfollows that under thehypothesesofTheorem2.3,h-c(E)
<h(g[-1](4cap(E)))
providedcap(E) <
g(-(-2))
Thisextends toarbitrarysetsEwithcap(E) <g(-14---2)
andwhencap(E) > g(1)/4,wemay simply scaleE by multiplyingall its elementsbysome smallpositivenumber. Similarly,Corollary2.4gives
h-c(E)
<h([4cap(E)] 1/A) (2.14)
andCorollary 2.5 givesforh(t)
(log )-’
F > 1h(4cap(E)). (2.15)
It
isstillpossibleto obtainnon-trivial estimateswhen theintegralin(2.2)
diverges. Onestillchoosesrj by(2.10),but instead estimatesI-I
rj >_ 4-(n-l)exp(I)j=2
where
)
I= loght-ll
Uh(H)
du= h(H)nfhI
1-11(h(H)/n)(log
v)h’ (v)
dr.On integrating by parts,weseethat the termcomingfromrlcancels,and we obtain
rj >4-nexp n logH gn
(H)
nj=
h(H)
[-ll(h(H)/n) 13 4and hence
In specificcasessuchash(t)
:= (log tl-) -
direct calculationofgn(H)leads to( ) {
(l+logn)h(43),n
1-’(1
y)-h(46), 0 <,
< 1(2.16)
While wehavefocusedonpolynomials, Can’slena has someofits mostpowerful
applicationswhenappliedtopotentials[8,11].Theguments e simil to that ofTheorem2.3,but the formulation is different.Let
us brieflyindicatethe extension togeneralizedpolynoals 1,6]. A
generalized polynomialofdegreenisanexpressionm m
P(z) H Iz
zjEJ
n.j=l j=l
Here allaj > 0 but are not necessarilyintegers. (Even n need notbe an integer,asthe reader willeasily
see).
All the estimates of Theorem 2.3 and itscorollariesgothroughfor such P. Indeed,because ofcontinuity,wecan assumethat allajarerational, andhaveformkj/N
for somepositive integerskj
and somepositive integer N independentof j. Then ifm
Q (z)
"=H (z zj)kJ
j=l
we seethat
Q
ismonicofdegreenN
andE<P; "IP<z)I a n} {z" Ia<z)l- nN}.
Since(2.4), (2.8),
(2.9)
do notdependon n,N,
theyremain validfor the more generalform ofP.However
thisshouldhardlybesurprisingasE(P;e)
still hascapacity e, so we are simply reformulating specialcases of(2.13)
to(2.15).
3
THE POLYA APPROACH
We
shall follow Goluzin[7]inproving P61ya’s THEOREM 3.1 Formonicpolynomials Pof
degreen,meas2(E(P; e))
< yre 2withequality
iff P(z)
(za) n.
Proof We
splitthis intoseveralsteps.STEP
1: Describe the lemniscater
.={z IP(z)l-
e}
as a union ofcontours
1-’j,
1 < j <m.Consider the map
P(z)
and its inverse algebraic functionz
p[-1]().
If1-" contains noneof thepointsz
withPt(z)
0,then 1-’ consists offinitely many disjointclosedanalyticJordan curves,say1-’j,
1 < j < m.By
the maximumprinciple,P (z)[
< eninsideeach1-’j
andeachI’j
encloses atleast one zero ofP.Thenm < n.STEP
2: Parametrizeeach1-’j.
Let
us suppose that1-’j
contains zeros of total multiplicitylj. As z
moves around1-’j, P(z)
moves aroundI1 n
exactlylj
times andP(z)l/b
moves once around the circleI1
/lj.Hence
one of thebranches of
z p[-1](lj)
is analytic and single valued onI1 n/b,
admittingthere the
Laurent
seriesexpansionp[-1](b)-- aJ)
ksothat for a suitable branch
p[-1]() aJ)k/b (3.2)
Thensetting l?neiO we obtain aparametrizationof
1-’j,
yj(O) :-- aJ)(enei) k/lj,
0 E [0,2rlj]. (3.3)
SxEe
3: Calculate areaenclosedbyFj
and hence the areaenclosedbyF.
A
well knownconsequence ofGreen’stheorem is aformulafortheareaAj
enclosedbyFj
lfr(xdy-ydx)=Im(frdz )
Aj -
where the secondintegralis acomplexcontourintegral. Usingourparametriza- tion(3.3),we seethat
Adding over j, we see that the area
A
enclosedby F admitstheidentityA
rn ;>n Ik
en/-j=l k=-c
(3.4) STEI"
4:Prove
P61ya’s inequality by lettinge--+ cxz.From
(3.4),
we seethat fork > 0,the termsintheserieshave positive coefficientsandpositive powers of e, increase with e, while the terms for k < 0havenegativecoefficientsandnegativepowersof e, also increase with e. ThusA/(zre 2)
increases with e. Butforlargee,F
consists ofonly one curveF1
andpl-1]has on the curveI1 gn
theexpansionp[-1]()
1/n-Jr"ao
(1) -k-a(_l
-1/n-t- a(l_)2-2/n
q-correspondingtothe inverse of the polynomial
P(z)
about cxz.(1) 1 and Integratingover
F1
gives as before(recall
now n and aa
) 0,k > 2),A _lail)]Zke2_z
1(1) ],a_-,a
y,2 k=-c k=l
Thus forlargee,A/(n’g
2) _<
1, andhence for all smallerealso.Moreover,
() 0, k > 1,that is, we seethat there isequalityiff
a_
p[-1]() l/n
’Jrao , z P(z)
1/n+ ao . P(z)
(zao) n.
COROLLARY3.2
For
bounded BorelsetsF, meas2F
<re(capF) 2.
Proof
IfF
iscompact,thengivene >0,we can(bydefinitionofcap)findnand a monicpolynomial Pofdegreen such that
F
CE(P;
cap(F)-t- ).
Then Theorem3.1 gives
meas2(F)
<meas2(E(P;
capF+ e))
< re(cap F+ e) 2.
Then
(3.5)
follows onlettinge --+Oh-.
Themeasurabilityandcapacitability ofgeneralBorel sets can be used to deduce thegeneralcase[8]. []We
noteonerelated result also due toP61ya, also provedabout the same time:IfPis a monicpolynomialofdegreen, andLisanyline intheplane, thenthere is thesharpestimatemeasl
(E(P; e)
fqL) <22-1/n. (3.6)
Here
measl denotes linear Lebesguemeasure. Theproofof this involves factorization ofP
and successively moving the intervals of E(P;e)
N L thereby showingthat the measure is maximized when E(P;e)
is a single intervaland P isessentially,aChebyshev polynomial ofdegreen. This is similarto aRemez
type argument,for those familiar with the latter.It has been conjectured that if we replace measl by one-dimensional Hausdorff measure, then
(3.6)
holds without having toprojectE(P;e)
onto itsintersection with the line L. If proved,thiswould show that forot 1, thesharpconstant inCartan’s(1.4)
is4,not4e.WhileP61ya’s proofabove is a beautifulapplicationof
Green’s
theorem,it seemsverycloselytied toplanarmeasure.It
would benice tosee theabove argument modified to treat other measures, if this is at allpossible.4
POLYNOMIALS WITH L
t,NORMALIZATION
Despite their beauty,
Cartan
andP61ya’s
estimates are closely linked to one variable: Factorization of polynomials in higher dimensions is more complicated, andGreen’s
theorem is very much a plane animal.Moreover,
there aremanynotionsof what constitutes amultivariate monic polynomial. So in the course ofinvestigating convergence of multivariatePad6approximants,
A. Cuyt, K.
Driverand the author were forced to consider alternative approaches, and the normalization(1.2)
was undoubtedly the easiest toextend.Tooursurprise,we discoveredsharpunivariateinequalities with this normalization, and moreover theproofsaresimple,thoughthese do involvethe notionofGreen’s
functions.In
this section, we shall presentsomeof these results for the univariate case,proved forp cxz in[5]
andfor 0 < p < cx in[16]. Let
us setP(rei)[PdO
IIPIIG(IzI-F)-
(1 f
exp -y log
[e(rei)[dO
0<p<c p=0
For
p cxz,the norm is as usualMoreover,
letusdefineto0 := 1;tc:--
2 andI /7r F (L2) ( +
1)1
0 < A < cx.tc 2
"--F-"
It
follows easily fromBeta
function identitiesthatxz
_< 216]. Moreover,
Stirling’sformulagivesxz
=2+
O(L_)
).
THEOREM4.1
Let
r, e > O andO
< p <_ cx. LetP
beapolynomialof
degreeat mostn, normalized bythe condition
Let
Then
IlellG(Izl=r)
1. (4.1)E(P;r;
s)"-- {z "]Zl
< r,[P(z)[
<en}. (4.2)
cap(E(P;
r;e))
< rStCnp;meas2(E(P;
r;e))
<yr(rStCnp) 2. (4.3)
These aresharpfor each r,nin the sense that
cap(E(P;
r;s)) meas2(E(P;
r;s))
sup rlCnp; sup
s>O t3 s>O
62
deg(P)--n deg(P)--n
7r(rtCnp) 2.
(4.4)
Proof LetussetE
"= E(P;r;e).
The result is trivial ifcap(E) 0,sowe assume that it ispositive.We
shall use some facts frompotential theory.The mostelementarytreatmentofthese appearsin[9], deepertreatmentsappear
in[3, 8, 11 ].The setE
haspiecewise analyticboundary,and isregularwith respecttothe Dirichletproblem.As
such it has a classicalGreen’sfunction g(z)withpoleatcx. Thishas thefollowing properties:g is harmonic inC\E;
g(z) log
Izl
/O(1), Izl ;
andg hasboundary
value0ontheboundaryofE.
Moreover,
g 0 inE. It
isknown that g admits therepresentation g(z) logIz tldx(t) +
logcap(E)
where
x
is aprobabilitymeasure withsupportonE,
the so-calledequilibrium measure ofE.For
polynomialsR
ofdegreem <n, there is the Bernstein- Walshinequality[e(z)[
<eg(Z)llell(E), z C. (4.5)
Theproofis simple:The functionF(z)
:= logIn (z) l-mg(z)-log R
is subharmonic in
C,
withboundaryvalue < 0 onE
and with a finite limit atee.The maximumprinciplefor subharmonic functionsgives F(z)
< 0 inC\E,
that is,(4.5)
followsas g > 0. Likewise onE,
theinequalityistrivial as g > 0.Now
inour caseR P
has maximumEnonE,
SOournormalization(4.1)
and(4.5)
give1
P [Itp(Izl--r)
<en eng(z) [[tp(Izl-r)
en
f
logIz-tldlz(t)cap(E)
Ilzp(Izl-r).
Let
us supposenow 0 < p < c. UsingJensen’s
inequality forintegrals appliedtothe convex function et,
we can continue this ascap(E)
enP f
logIrei-tldlx(t)dO
f
< cap(E)
Ire
iOtlnpdlz(t)
dO=(ca(E))[f[2fo2lrei-tlnpdO]dtz(t)l
1/p<
ca12(e)
[tl<rsup- Irei tlnpdo (4.6)
In
the second last line, we used Fubini’stheorem, and in the last line, we used the factthat/z
hassupportinE
C{z [zl
<_r}.
Itis clearthatrotating does notchangethe value of theintegralinthesupsowemaytake 6 [0, r].Moreover,
thefact that theLp
normof ananalyticfunction on a circle centre 0, radius t, increases with [20, p. 337] givesthatthe sup is attained forr, so
1 <_
cap(E)
[e
iOl[npdo
cap(E) cos dO
Expressingthe lastintegralintermsof
Beta
functionsgives rEKnptn
and hence we have the firstinequalityin
(4.3).
P61ya’s inequalityTheorem 3.1 thengivesthe secondinequalityin(4.3).
The case p cx is easier, as we seethatthesup in(4.6)
thenbecomes(2r) n.
The case p 0requires morecare, see 16].To
prove the sharpness for0 < p < cx, we let 0 < a < r, andP(z)
"=(z___a)n,
where) is chosen to give the normalization(4.1). It
is easytoseethat for smallenoughe, E(P;r;e) {z Iz al
_<e)}
and someas2(E(P;
r;e)) ,2 cap(E(P;
r;e))
); re
E 62
The normalization
(4.1)
shows that- IreiO alnpdo -- (rlCnp) n,
a--+ r.
So)maybe madearbitrarilyclosetoKnp. Then
(4.4)
follows.We
note onegeneralization,provedin[16]:
THEOREM 4.2 with
Let
p
(0,x)
be strictly increasing andcontinuous7t(0)
:-- lim ap(t) < 1 < lim ap(t)=:(o).
t-+0+ t--+c
Assume,
moreover, that gr(et)
is convex in (-o,o).
Let r, e > 0 and 0 < p < o. LetP beapolynomialof
degree at mostn, normalized by the condition2--1 fo
2zrP\,P(rei),P/dO(I
1.(4.7)
Let
tcz,O be the rootof
the equation--2zr
1
f02r ([ll-eil]
z dO=l.Then theestimates
(4.3)
holdif
wereplaceXnp byXnp,/andmoreoverthese aresharpin thesensethat(4.4)
holds with Knp replaced byKnp,O.Theproofisverysimilar tothat of Theorem4.1,onejust applies
Jensen’s
inequality toq/(et)
rather than et.
See[16]
for this and further extensions involving generalized polynomialsandpotentials.5
REMEZ
INEQUALITIESRemez
inequalitieshave been studiedintensively byTamas
Erdelyiand his collaborators inrecentyears.Theyhave been shown to be useful inproving Markov-Bernsteinand Nikolskiiinequalities, amongstothers. The classical one involvestheChebyshev polynomialTn (x):
THEOREM 5.1
Let P
be a polynomialof
degree at most n, with realcoefficients,
and( "=measl
{x
[-1,1]"IP(x)l
<1}. (5.1)
Then
,IPIIL[_I,I]
<Tn (-- I). (5.2)
Thereisequality
iff P(x) -+-Tn (+2x-2-).
Thereadermayfindanelegantproofinthedelightfulbook of Borwein and Erdelyi [1]:
As
withP61ya’s(3.6),
theproofinvolves shiftingthe intervals thatcomprise{x
[-1, 1]"P (x)
<1}
untilonehas asingleinterval, at eachstageincreasingthe measure.For
ourpurpose, thefollowingcorollary isof most interest: Recall thatTn
isstrictly increasingin [1,oe)
withTn (1)
1.Hence
ithasaninverseTn[-1]
[1,o)
--+ [1,cx).COROLLARY 5.2 Let
P
be a polynomialof
degree at most n, with realcoefficients,
normalized byIIPIIzoo[-1,1]
1.(5.3)
Let
e (0, 1].Then4
measl
{x
[--1,1]Ie(x)l
< 6n}
< (<_23-1/n6). (5.4)
1
+ Zn[-1](e -n)
Given e 6 (0, 1], we have equality in
(5.4)for
suitablepolynomialsP of
degreen.
Note
thepowerof thecorollary: Itissharpfor each e, notjustase --+0+.
Proof of
Corollary5.2For
thegivenP,
letQ
:=e-nP.Then{x I-a, 11" Ie(x)l {x
[-1,1].[Q(x)
_<1}
=. E.By Remez’s
inequality(5.2),
(4
Ilell o t-l,la
< 1measlE
and our normalization
(5.3)
givese-n
<Tn( measE
4-1).
Inverting thisgives
(5.4).
The secondinequality in(5.4)
follows from the elementary inequalityrn(x)
<_2n-lxn,
x [1,o)== Tn[-1](u)
>2-1+l/nul/n,
u [1,(5.5)
ThesharpnessisasimpleconsequenceofthesharpnessofTheorem 5.1: Fix e and set4
()
:= =Tn
-1 =e-n.
1
+ rnt-ll(e -)
Theorem5.1 shows that
P (x) -Fe
nTn
+2x+e-)
gives equalityin(5.4).rn We
emphasisethatRemez inequalities have beenprovedforgeneralized polynomials, potentials,inLp
spaces,for Miantzpolynomials See [1,6]
forresults and references.Sinceouremphasisis onordinary polynomials,and regionsintheplane,we restrictourselves to thefollowingresult ofErdelyi, Li and Saff[6,Theorem
2.5]
for the unitball:THEOREM 5.3 with
Lets [0, 1/4]
and P beapolynomialof
degreeatmostnThen
Here
Cisindependentof
n,P,
s.What is fairly typicalabout this
Remez
extension is that itapplies onlywhenmeas2(E(P;
1;1))isboundedawayfrom0,namelywhenitis
>zr-.
This is indicative of the rationale of Remez inequalities. Their greatest use is when
meas2(E(P;
1;1))
approaches its full measure zr, while the inequalitiesofthe previoussections are most useful whenmeas2(E
(P; 1; 1))
approaches0.
6
MULTIVARIATE POLYNOMIALS
The polynomial P(Zl, Z2) :-" (ZlZ2)n illustrates many of the multivariate features. The lemniscate
E<e;
"=lP<z ,z=>l {(Zl,Z2). lZlZ2l
is unbounded and even has infinite
(4
dimensional) Lebesgue measure.Moreover,
is the degree ofP,
n or 2n?We
shall define its degree as n.We
shallsaythat thedegreeof apolynomial P(z
l,z2 z)ofkvariables is n, ifthehighest powerof eachzj is atmost n, withequalityforatleast one j. Totake account of the unboundedness ofE(P; 8), we considerthe restrictedlemniscateE(P;r;
8):-- {(Zl,Z2
Zk)[Zjl <_
rVj,IP(zl,z2
Zk) <_8n}.
A. Cuyt, K.
Driver andtheauthor[5]
used Theorem 4.1andinductiononk toprove:THEOREM 6.1 normalized by
Letr, e > O. Let P be apolynomial
of
degreeat mostn,max{]P(z,z2 z)l" IzjI
_<rj]--1.
Then
if
meas2kdenotes Lebesguemeasure inC
k(=N2),
(6.1)
2k_ }k-1
meas2(E(P;
r;e))
<(16yrr2)e2
max 1, log2(6.2)
While the constants are not sharp, the powers of r, e are, including the(ZlZ2/r2)
surprisinga calculationfactor log2shows that. For
k 2 and the polynomial P(zl,z2)
meas2(E(P;r;e))-- (yrr2)2e2 [1 +
2log 11
What aboutmonic normalization? B. Paneah
[17]
generalized Cartan’s lemma as follows. Given a multi-index (oil,or2 Otk)wesaythat it is leading if(I)
0aP:=""\Oz]
P#O;
(II)
1 5j5ksuchthatj 0,
""kOzl
P 0"For example,
P (z
z2 z3z + z + z + 3z ZzZ3
2 2 hasleadingmulti-indices (3, 0, 0), (0, 3,0)
and(0, 0,2)
butnot(2, 2,1).
For
(Zl,z2 z)and 1 5 j 5 k,wesetj
"= (Zl,z2 zj-,zj+,z). Onedimensional lines in
C
palleltothezj axishavethe fowherea6
C
k-1 Paneahproved:THEOREM6.2
Let P
beof
degree n, and be a leading multi-index. Let8j
> 0 withequalityiff
jO,
1 < j < k. Thereexistsubsetsjk/lj of C
ksuch that
Moreover, Mj
intersectsany lineCj (a),
aC
k-1,
inatmostjcircles, with sumof
diameters <43j.
Note
thatfor k 1,ot nand we obtain(31 / n)n
n!,
preciselythequantity intheproofofCorollary2.2.What about measures other thanLebesgue
measure?
Notions ofcapacity are farmorecomplicatedinthe multivariate case, and several basicquestions remain unresolved. One of the main problems is the lack of an explicit formula fortheGreens’function.See [2, 5, 10, 12, 13,22]
for partial results.Undoubtedlythegreatestscopeforworkonsmallvalues ofpolynomialslies inthe multivariatesetting.
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