Table Of ContentGENERALIZATIONS OF GONC¸ALVES’ INEQUALITY
5
0
PETERBORWEIN,MICHAELJ.MOSSINGHOFF,ANDJEFFREYD.VAALER
0
2
n Abstract. LetF(z)= Nn=0anznbeapolynomialwithcomplexcoefficients
a and roots α1, ..., αN, lPetkFkp denote its Lp norm over the unit circle, and
J letkFk0 denoteMahler’smeasureofF. Gonc¸alves’ inequalityassertsthat
1 N N 1/2
1 kFk2≥|aN| max{1,|αn|2}+ min{1,|αn|2}
n=1 n=1 !
Y Y
] 2 1/2
CA =kFk0 1+ |ak0FaNk40| ! .
Weprovethat
.
h N N 1/p
at kFkp≥Bp|aN| max{1,|αn|p}+ min{1,|αn|p}
m n=1 n=1 !
Y Y
[ for1≤p≤2,whereBp isanexplicitconstant, andthat
v1 kFkp≥kFk0 1+ p24|ak0FakN40|2!1/p
3 for p ≥ 1. We also establish additional lower bounds on the Lp norms of a
6 polynomialintermsofitscoefficients.
1
1
0
5
0 1. Introduction
/
h Let∆ Cdenotetheopenunitdisc,∆itsclosure,andlet (∆)denotethealge-
at braofcon⊂tinuous functionsf :∆ C thatare analyticon∆A. Then (∆),
m is a Banach algebra,where → {A k·k∞}
:
v f =sup f(z) :z ∆ =sup f(e(t)) :t R/Z ,
k k∞ | | ∈ {| | ∈ }
i
X and e(t) denotes the func(cid:8)tion e2πit. If f(cid:9) (∆) and 0<p< , we also define
∈A ∞
r
a 1 1/p
f = f(e(t))pdt ,
k kp | |
(cid:18)Z0 (cid:19)
and we define
1
f =exp log f(e(t)) dt .
k k0 | |
(cid:18)Z0 (cid:19)
It is known (see [2, Chapter 6]) that for each f in (∆) the function p f is
A → k kp
continuous on [0, ], and if 0<p<q < , then these quantities satisfy the basic
∞ ∞
2000 MathematicsSubjectClassification. Primary: 30A10,30C10;Secondary: 26D05,42A05.
Key words and phrases. Lp norm, polynomial, Gonc¸alves’ inequality, Hausdorff-Young
inequality.
ResearchofP.BorweinsupportedinpartbyNSERCofCanadaandMITACS.
1
2 PETERBORWEIN,MICHAELJ.MOSSINGHOFF,ANDJEFFREYD.VAALER
inequality
(1.1) f f f f .
k k0 ≤k kp ≤k kq ≤k k∞
Clearly, equality can occur throughout (1.1) if f is constant. On the other hand,
if f is not constant in (∆), then the function p f is strictly increasing on
A → k kp
[0, ].
N∞ow suppose that F is a polynomial in C[z] of degree N 1, and write
≥
N N
(1.2) F(z)= a zn =a (z α ).
n N n
−
n=0 n=1
X Y
In this case, the quantity F is Mahler’s measure of F, and by Jensen’s formula
k k0
one obtains the well-known identity
N
(1.3) F = a max 1, α .
k k0 | N| { | n|}
n=1
Y
Thus a special case of (1.1) is the inequality (often called Landau’s inequality)
N
F a max 1, α .
k k2 ≥| N| { | n|}
n=1
Y
For polynomials of positive degree, the sharper inequality
1/2
N N
(1.4) F a max 1, α 2 + min 1, α 2
k k2 ≥| N| { | n| } { | n| }!
n=1 n=1
Y Y
was obtained by Gonc¸alves [1]. Note that equality occurs in (1.4) for constant
multiples of zN 1. Alternatively, the inequality (1.4) may be written in the less
−
symmetrical form
1/2
a a 2
0 N
(1.5) F F 1+ | | .
k k2 ≥k k0 F 4 !
k k0
For a positive real number p, define the real number B by
p
1 1 1/p Γ(p+1) 1/p
(1.6) B = 1 e(t)p dt = ,
p 2 | − | 2Γ(p/2+1)2
(cid:18) Z0 (cid:19) (cid:18) (cid:19)
and note that B = 2/π and B = 1. In this article we establish the following
1 2
generalizations of Gonc¸alves’ inequality.
Theorem 1. Let F(z) C[z] be given by (1.2). If 1 p 2, then
∈ ≤ ≤
1/p
N N
(1.7) F B a max 1, α p + min 1, α p
k kp ≥ p| N| { | n| } { | n| }!
n=1 n=1
Y Y
and if p 1, then
≥
1/p
p2 a a 2
0 N
(1.8) F F 1+ | | .
k kp ≥k k0 4 F 4 !
k k0
GENERALIZATIONS OF GONC¸ALVES’ INEQUALITY 3
Equality occurs in (1.7) for constant multiples of zN 1. The inequality (1.8)
−
is never sharp for p =2, but since B <1 for 1 p <2 it is clearly stronger than
p
6 ≤
(1.7) in this range when F 2/ a a is large. For example, one may verify that
k k0 | 0 N|
(1.8) produces a better bound in the case p=1 whenever
F 2 π2
k k0 > =1.1576382...
a0aN 2 2 √4+2π π2
| | − −
Also,forfixedF therightside(cid:0)of(1.8)achievesa(cid:1)maximumatp=2c F 2/ a a ,
k k0 | 0 N|
where c = 1.9802913... is the unique positive number satisfying 2c2 = (1 +
c2)log(1+c2). In view of (1.1), inequality (1.8) is therefore only of interest when
1 p 2c F 2/ a a .
≤ ≤ k k0 | 0 N|
To prove Theorem 1, we first establish some lower bounds on the L norms of
p
a polynomial in terms of two of its coefficients a and a , provided M L is
L M
| − |
sufficientlylarge. Theseinequalitieshavesomeindependentinterest,andwerecord
the results in the following theorem.
Theorem 2. Let F(z) C[z] be given by (1.2), and let L and M be integers
∈
satisfying 0 L<M N and M L>max L,N M . Then
≤ ≤ − { − }
(1.9) F a + a .
k k∞ ≥| L| | M|
Further, if 1 p 2 then
≤ ≤
(1.10) F B (a p+ a p)1/p,
k kp ≥ p | L| | M|
and if p 1 and a and a are not both 0, then
L M
≥
2 1/p
pmin a , a
L M
(1.11) F max a , a 1+ {| | | |} .
k kp ≥ {| L| | M|} (cid:18)2max{|aL|,|aM|}(cid:19) !
At this point, it is instructive to recall the Hausdorff-Young inequality. If p=2
and F(z) is given by (1.2), then by Parseval’s identity we have
1/2
(1.12) F = a 2+ a 2+ + a 2 .
k k2 | 0| | 1| ··· | N|
(cid:16) (cid:17)
If p=1, then the inequality
(1.13) F max a , a ,..., a
k k1 ≥ {| 0| | 1| | N|}
follows immediately from the identity
1
a = F(e(t))e( nt)dt.
n
−
Z0
Now suppose that 1 < p < 2 and let q be the conjugate exponent for p, so p−1+
q−1 =1. Then the Hausdorff-Young inequality [3, p. 123] asserts that
(1.14) F (a q+ a q + + a q)1/q,
k kp ≥ | 0| | 1| ··· | N|
and so interpolates between (1.12) and (1.13). If p = 2, then (1.10) and (1.11)
are equivalent and clearly follow from the identity (1.12). But for 1 < p < 2, the
inequalities (1.10) and (1.11) are not immediate consequences of (1.14). In fact, it
iseasytoseethatthelowerboundsin(1.10),(1.11),and(1.14)arenotcomparable.
If p=1, the same remarks apply to (1.10), (1.11), and (1.13).
4 PETERBORWEIN,MICHAELJ.MOSSINGHOFF,ANDJEFFREYD.VAALER
In section 2 we develop some preliminary results concerning lower bounds on
L norms of binomials, and we use these facts to establish Theorems 1 and 2 in
p
section 3.
2. Norms of binomials
For 0<r <1 and real t, recall that the Poisson kernel is defined by
∞ 1+re(t) 1 r2
P(r,t)= r|n|e(nt)= = − .
ℜ 1 re(t) 1 re(t)2
n=X−∞ (cid:18) − (cid:19) | − |
This is a positive summability kernel that satisfies
1
P(r,t)dt=1
Z0
and
1−ǫ
lim P(r,t)dt=0
r→1−
Zǫ
for 0<ǫ<1/2.
Lemma 3. If p>0 then
1
lim 1 re(t)pP(r,t)dt=0.
r→1−Z0 | − |
Proof. Let 0<ǫ<1/2 so that
1 ǫ 1−ǫ
1 re(t)pP(r,t)dt= 1 re(t)pP(r,t)dt+ 1 re(t)pP(r,t)dt
| − | | − | | − |
Z0 Z−ǫ Zǫ
ǫ 1−ǫ
1 re(ǫ)p P(r,t)dt+2p P(r,t)dt
≤| − |
Z−ǫ Zǫ
1−ǫ
1 re(ǫ)p+2p P(r,t)dt.
≤| − |
Zǫ
We conclude that
1
limsup 1 re(t)pP(r,t)dt 1 e(ǫ)p (2πǫ)p,
| − | ≤| − | ≤
r→1− Z0
and the statement follows. (cid:3)
For positive numbers p and r, we define
1 p
(2.1) (r)= 1 r1/pe(t) dt.
p
I −
Z0 (cid:12) (cid:12)
It follows easily that r (r) is a co(cid:12)ntinuous, po(cid:12)sitive, real-valued function that
→Ip (cid:12) (cid:12)
satisfies the functional equation
(2.2) (r)=r (1/r)
p p
I I
forallpositiver. Thefollowinglemmarecordssomefurtherinformationaboutthis
function.
Lemma4. Foranypositivenumberp,thefunctionr (r) has acontinuousde-
p
→I
rivativeateachpointof(0, )andsatisfiestheidentity ′(1)= (1)/2. Moreover,
∞ Ip Ip
this function has infinitely many continuous derivatives on the open subintervals
(0,1) and (1, ).
∞
GENERALIZATIONS OF GONC¸ALVES’ INEQUALITY 5
Proof. Suppose first that 0 < r < 1. Then 1 r1/pe(t) p/2 has the absolutely
−
convergentFourier expansion
(cid:0) (cid:1)
p/2 p/2
1 r1/pe(t) = ( 1)mrm/pe(mt).
− m −
(cid:16) (cid:17) mX≥0(cid:18) (cid:19)
By Parseval’s identity, we have
2
p/2
(2.3) (r)= r2m/p.
p
I m
m≥0(cid:18) (cid:19)
X
This shows that r (r) is represented on (0,1) by a convergent power series in
p
→I
r1/p andthereforehasinfinitelymanycontinuousderivativesonthisinterval. Next,
we observe that
∂ p 1 r1/pe(t) p
1 r1/pe(t) = − 1 P(r1/p,t) .
∂r − 2r −
(cid:12) (cid:12)
Itfollowsthatif0(cid:12)(cid:12)<ǫ 1/4an(cid:12)(cid:12)dǫ (cid:12)r 1 ǫ,th(cid:12)en(cid:16)thereexistsa(cid:17)positiveconstant
(cid:12) ≤ (cid:12) ≤ ≤ −
C(ǫ,p) such that
∂ p
1 r1/pe(t) C(ǫ,p).
∂r − ≤
tFhraotm the mean value theo(cid:12)(cid:12)(cid:12)(cid:12)rem(cid:12)(cid:12)(cid:12)and the do(cid:12)(cid:12)(cid:12)m(cid:12)(cid:12)(cid:12)(cid:12)inated convergence theorem, we find
1 1 p
′(r)= 1 r1/pe(t) 1 P(r1/p,t) dt,
Ip 2r − −
Z0 (cid:12) (cid:12) (cid:16) (cid:17)
and therefore (cid:12) (cid:12)
(cid:12) (cid:12)
1 p
(r) 2r ′(r)= 1 r1/pe(t) P(r1/p,t)dt.
Ip − Ip −
Z0 (cid:12) (cid:12)
Using the continuity of r (r) and(cid:12)Lemma 3, w(cid:12)e conclude that
→Ip (cid:12) (cid:12)
(2.4) lim ′(r)= (1)/2.
r→1−Ip Ip
Again using the mean value theorem, it follows that r (r) has a left-hand
p
→ I
derivative at 1 with the value (1)/2.
p
I
From (2.2) and (2.3) we find that
2
p/2
(2.5) (r)=r r−2m/p
p
I m
m≥0(cid:18) (cid:19)
X
for r > 1. Thus r (r) is represented by r times a convergent power series
p
→ I
in r−1/p, and so has infinitely many continuous derivatives on the interval (1, ).
∞
Next, we differentiate both sides of (2.2) to obtain the identity
′(1/r)
′(r)= (1/r) Ip
Ip Ip − r
for r>1, and using the continuity of r (r) and (2.4), we conclude that
p
→I
lim ′(r)= (1) lim ′(s)= (1)/2.
r→1+Ip Ip −s→1−Ip Ip
It follows that r (r) has a right-hand derivative at 1 with value (1)/2.
p p
→I I
We conclude then that r (r) is continuously differentiable on (0, ) and
p
′(1)= (1)/2. → I ∞ (cid:3)
Ip Ip
6 PETERBORWEIN,MICHAELJ.MOSSINGHOFF,ANDJEFFREYD.VAALER
From the proof of the lemma we obtain the following lower bound on the L
p
norm of a binomial.
Corollary 5. Let 0 L<M be integers and let α and β be complex numbers, not
≤
both zero. If p>0 then
pmin α , β 2 1/p
αzL+βzM max α , β 1+ {| | | |} ,
p ≥ {| | | |} (cid:18)2max{|α|,|β|}(cid:19) !
(cid:13) (cid:13)
with equal(cid:13)ity precisely(cid:13)when αβ =0 or p=2.
Proof. The result is trivial if either α or β is zero, so we assume that this is not
the case. We maythen assumeby homogeneitythatα=1, anditis clearfromthe
definitionof f that we may assume that L=0,and thatβ is realand negative.
k kp
If β <1, then taking r = β p in (2.3) and keeping just the first two terms of the
| | | |
sum, we obtain
1+βzM p = 1+βz p 1+ p2|β|2.
p k kp ≥ 4
If β >1, then (cid:13) (cid:13)
| | (cid:13) (cid:13)
1+βzM p = β p β−1+z p = β p 1+z/β p,
p | | p | | k kp
so taking r= β(cid:13)−p, we ob(cid:13)tain in th(cid:13)e same w(cid:13)ay
(cid:13) (cid:13) (cid:13) (cid:13)
| |
1+βzM p β p 1+ p2 .
p ≥| | 4 β 2!
| |
(cid:13) (cid:13)
The case β = 1 follow(cid:13)s by conti(cid:13)nuity. For the case of equality, notice that the
sum (2.3) has p−recisely two nonzero terms only when p=2. (cid:3)
The next lower bound is obtained by establishing the convexity of the function
r (r) for each fixed p in (0,2].
p
→I
Lemma 6. If 0<p 2, then the function r (r) satisfies the inequality
p
≤ →I
(1)(1+r)
p
(2.6) (r) I
p
I ≥ 2
for r >0.
Proof. If p = 2 then (r) = 1+r and the result is trivial. Suppose then that
2
I
0 < p < 2. If r < 1, then we may differentiate the power series (2.3) termwise to
obtain
2
p/2 2m
′(r)= r(2m/p)−1.
Ip m p
m≥0(cid:18) (cid:19)
X
As 0<p<2, it follows that r ′(r) is strictly increasing on (0,1), so r (r)
→Ip →Ip
is strictly convex on this interval. Thus, if r and s are in (0,1), then
(2.7) (r) (s)+(r s) ′(s).
Ip ≥Ip − Ip
Letting s 1 and using Lemma 4, we obtain
→ −
(1)(1+r)
(2.8) ′(r) (1)+ ′(1)(r 1)= Ip .
Ip ≥Ip Ip − 2
for 0<r <1.
GENERALIZATIONS OF GONC¸ALVES’ INEQUALITY 7
In a similar manner, if r >1 we differentiate (2.5) termwise to obtain
2
p/2 2m
′(r)= 1 r−2m/p,
Ip m − p
m≥0(cid:18) (cid:19) (cid:18) (cid:19)
X
andagainr ′(r) is strictly increasingon(1, ), so r (r) is strictly convex
→Ip ∞ →Ip
on this interval. Thus (2.7) holds as well for r > 1 and s > 1, and letting s 1+
→
we obtain (2.8) for r>1.
We havethereforeverified(2.6)ateachpointr in(0,1) (1, ), anditistrivial
at r =1. ∪ ∞ (cid:3)
Usingthislemma,weobtainasecondlowerboundontheL normofabinomial.
p
Corollary 7. Let 0 L<M be integers and let α and β be complex numbers. If
≤
0<p 2 then
≤
αzL+βzM B (αp+ β p)1/p.
p ≥ p | | | |
Proof. Theresultistri(cid:13)vialifeither(cid:13)αorβ iszero,soweassumethatthis isnotthe
(cid:13) (cid:13)
case. By homogeneity, we may assume then that α =1, and we may assume that
| |
L = 0 and that β is real and negative by the definition of f . Using Lemma 6,
k kp
we obtain
1+βzM p = 1+βz p
p k kp
(cid:13) (cid:13) = p(β p)
(cid:13) (cid:13) I | |
(1)(1+ β p)
p
I | |
≥ 2
=Bp(1+ β p).
p | |
(cid:3)
3. Proofs of the theorems
The proof of Theorem 2 employs an averaging argument and makes use of the
triangle inequality for L norms. We therefore require the restriction p 1 in the
p
≥
statement of the theorem.
Proof of Theorem 2. Suppose that F(z) = N a zn is a polynomial with com-
n=0 n
plex coefficients, L and M are as in the statement of the theorem, and 1 p 2.
Set K =M L, and let ζ denote a primitPive Kth root of unity in C. Th≤en ≤
K
−
K N K
1 1
ζ−kLF ζkz = ζk(n−L) a zn
K K K K K n
!
k=1 n=0 k=1
X (cid:0) (cid:1) X X
= a zn
n
0≤n≤N
X
n≡L(modK)
=a zL+a zM.
L M
Using the triangle inequality and the fact that the polynomials ζ−kLF(ζkz) all
K K
have the same L norm, we find that
p
K
1
(3.1) F ζ−kLF ζkz = a zL+a zM
k kp ≥(cid:13)K K K (cid:13) L M p
(cid:13)(cid:13)(cid:13) kX=1 (cid:0) (cid:1)(cid:13)(cid:13)(cid:13)p (cid:13)(cid:13) (cid:13)(cid:13)
(cid:13) (cid:13)
8 PETERBORWEIN,MICHAELJ.MOSSINGHOFF,ANDJEFFREYD.VAALER
for 1 p . The inequality (1.9) then follows by selecting a complex number
≤ ≤ ∞
z of unit modulus so that a zL and a zM have the same argument. Then in-
L M
equalities(1.10)and(1.11)areestablishedbycombining(3.1)withCorollary5and
Corollary 7, respectively. (cid:3)
TheproofofTheorem1proceedsbyapplyingTheorem2toapolynomialhaving
the same values over the unit circle as the given polynomial F. Ostrowski [6] and
Mignotte [5] (see also [4, p. 80]) employ a similar construction in their proofs of
Gonc¸alves’ inequality (1.5) in the case p=2.
Proof of Theorem 1. Suppose that F(z) = N a zn = a N (z α ) is a
n=0 n N n=1 − n
polynomial with complex coefficients. If F(z) has a root at z = 0, then (1.7)
P Q
and (1.8) follow immediately from (1.1), so we assume that a = 0. Let denote
0
6 E
the collectionof all subsets of 1,2,...,N , and for each E in , let E′ denote the
{ } E
complementofE in 1,2,...,N . ForeachsetE in ,wedefinethefiniteBlaschke
{ } E
product B (z) by
E
1 α z
n
B (z)= −
E
z α
n
n∈E −
Y
and the polynomial G (z) by
E
N
G (z)=B (z)F(z)= b (E)zn.
E E n
n=0
X
Clearly,
b (E)=a ( α )
0 N m
−
m∈E′
Y
and
b (E)=a ( α ).
N N n
−
n∈E
Y
If z =1 then the Blaschke product satisfies B (z) =1, so G (z) = F for
| | | E | k E kp k kp
0 p and every E in . Now select L = 0 and M = N for the polynomial
≤ ≤ ∞ E
G (z) in Theorem 2. Then from (1.10) we obtain
E
1/p
(3.2) F B a α p+ α p
k kp ≥ p| N| m∈E′| m| n∈E| n| !
Y Y
for1 p 2. Also,assuming withoutloss ofgeneralitythat b (E) b (E), we
0 N
≤ ≤ | |≥| |
find from (1.11) that
1/p
p2 b (E)2
N
F b (E) 1+ | |
k kp ≥| 0 | 4 b0(E)2 !
(3.3) | |
1/p
p2 a a 2
0 N
= b (E) 1+ | |
| 0 | 4 b0(E)4!
| |
for p 1. Inequalities (1.7) and (1.8) then follow from (3.2) and (3.3) by choosing
E = ≥n: α 1 . (cid:3)
n
{ | |≤ }
GENERALIZATIONS OF GONC¸ALVES’ INEQUALITY 9
WeremarkthatthechoiceofE intheprecedingproofproducesthebestpossible
inequality in (3.2). To establish this, suppose that E has b (E) = F /r
∈ E | 0 | k k0
for some real number r, so b (E) = r a a / F . Then certainly 1 r
| N | | 0 N| k k0 ≤ ≤
F 2/ a a , and it is easy to check that
k k0 | 0 N|
F r a a a a
k k0 + | 0 N| F + | 0 N|
r F ≤k k0 F
k k0 k k0
in this range, with equality occurring only at the endpoints.
It is possible, however, that a different choice for E in (3.3) could produce a
bound better than (1.8) for a particular polynomial. Specifically, if E has
∈ E
b (E) = F /r, again with 1 r F 2/ a a , then we obtain an improved
| 0 | k k0 ≤ ≤ k k0 | 0 N|
bound whenever
4 F 4+p2 a a 2 rp <4 F 4+p2 a a 2r4,
k k0 | 0 N| k k0 | 0 N|
and this may oc(cid:16)cur when p is small.(cid:17)For example, the polynomial F(z) = 18z2
−
101z+90 has roots α = 9/2 and α = 10/9; choosing E = with p = 1 yields
1 2
{}
F 90.9, but selecting E = 1 (so r = 9/2) produces a lower bound slightly
k k1 ≥ { }
larger than 102.
References
[1] J. V. Gonc¸alves, L’in´egalit´e de W. Specht, Univ. Lisboa Revista Fac. Ci. A (2) 1 (1950),
167–171. MR12,605j
[2] G.H.Hardy,J.E.Littlewood,andG.Po´lya,Inequalities,CambridgeUniv.Press,Cambridge,
1988.
[3] Y.Katznelson,Anintroductiontoharmonicanalysis,3rded.,CambridgeUniv.Press,Cam-
bridge,2004.
[4] M.MignotteandD.S¸tefa˘nescu,Polynomials:analgorithmicapproach,Springer-Verlag,Sin-
gapore,1999.
[5] M.Mignotte,Aninequalityaboutfactorsofpolynomials,Math.Comp.28(1974),1153–1157.
MR50#7102
[6] A. M. Ostrowski, On an inequality of J. Vicente Gon¸calves, Univ. Lisboa Revista Fac. Ci.
A(2)8(1960), 115–119. MR26#2585
DepartmentofMathematicsandStatistics,SimonFraser University,Burnaby,B.C.
V5A1S6Canada
E-mail address: [email protected]
DepartmentofMathematics,DavidsonCollege,Davidson,NorthCarolina28035USA
E-mail address: [email protected]
Departmentof Mathematics,University ofTexas, Austin, Texas78712USA
E-mail address: [email protected]
