INTEGRAL MEAN ESTIMATES FOR POLYNOMIALS
WHOSE ZEROS ARE WITHIN A CIRCLE
K.K. DEWAN, N. SINGH, B. CHANAM
Department of Mathematics
Faculty of Natural Sciences,
Jamia Millia Islamia (Central University)
New Delhi-110025, India
Integral Mean Estimates
K.K. Dewan, N. Singh,
B. Chanam and A. Mir
vol. 10, iss. 1, art. 23, 2009
Title Page
EMail: nareshkuntal@yahoo.co.in
ABDULLAH MIR
Post Graduate Department of Mathematics
University of Kashmir
Hazratbal, Srinagar-190006, India
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Received:
17 October, 2007
Accepted:
19 December, 2008
Communicated by:
G.V. Milovanović
2000 AMS Sub. Class.:
30A10, 30C10, 30D15
Key words:
Polynomials, Zeros of order m, Inequalities, Polar derivatives.
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Abstract:
Let p(z) be a polynomial of degree n. Zygmund [11] has shown that for
s≥1
Z
0
2π
|p0 (eiθ )|s dθ
1/s
Z
≤n
2π
|p(eiθ )|s dθ
1/s
.
0
In this paper, we have obtained inequalities in the reverse direction for the
polynomials having a zero of order m at the origin. We also consider a
n
P
an−ν z n−ν not
problem for the class of polynomials p(z) = an z n +
ν=µ
vanishing outside the disk |z| < k, k ≤ 1 and obtain a result which,
besides yielding some interesting results as corollaries, includes a result
due to Aziz and Shah [Indian J. Pure Appl. Math., 28 (1997), 1413–1419]
as a special case.
Integral Mean Estimates
K.K. Dewan, N. Singh,
B. Chanam and A. Mir
vol. 10, iss. 1, art. 23, 2009
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Acknowledgment:
The work of second author is supported by Council of Scientific and Industrial Research, New Delhi, under grant F.No.9/466(78)/2004-EMR-I.
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Contents
1
Introduction and Statement of Results
4
2
Lemmas
9
3
Proofs of The Theorems
10
Integral Mean Estimates
K.K. Dewan, N. Singh,
B. Chanam and A. Mir
vol. 10, iss. 1, art. 23, 2009
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1.
Introduction and Statement of Results
Let p(z) be a polynomial of degree n and p0 (z) its derivative. It was shown by Turán
[10] that if p(z) has all its zeros in |z| ≤ 1, then
n
max |p(z)| .
2 |z|=1
max |p0 (z)| ≥
(1.1)
|z|=1
More generally, if the polynomial p(z) has all its zeros in |z| ≤ k, k ≤ 1, it was
proved by Malik [5] that the inequality (1.1) can be replaced by
n
(1.2)
max |p0 (z)| ≥
max |p(z)| .
|z|=1
1 + k |z|=1
Malik [6] obtained a Lp analogue of (1.1) by proving that if p(z) has all its zeros in
|z| ≤ 1, then for each r > 0
(1.3)
r1
2π
Z
iθ
r
|p(e )| dθ
n
2π
Z
iθ r
≤
r1
|1 + e | dθ
0
max |p0 (z)|.
|z|=1
0
As an extension of (1.3) and a generalization of (1.2), Aziz [1] proved that if p(z)
has all its zeros in |z| ≤ k, k ≤ 1, then for each r > 0
2π
Z
(1.4)
|p(eiθ )|r dθ
n
0
r1
Z
≤
2π
|1 + keiθ |r dθ
0
r1
max |p0 (z)|.
|z|=1
If we let r → ∞ in (1.3) and (1.4) and make use of the well known fact from
analysis (see for example [8, p. 73] or [9, p. 91]) that
Z 2π
r1
→ max p(eiθ )
as r → ∞ ,
(1.5)
|p(eiθ )|r dθ
0
0≤θ<2π
Integral Mean Estimates
K.K. Dewan, N. Singh,
B. Chanam and A. Mir
vol. 10, iss. 1, art. 23, 2009
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we get inequalities (1.1) and (1.2) respectively.
In this paper, we will first obtain a Zygmund [11] type integral inequality, but in
the reverse direction, for polynomials having a zero of order m at the origin. More
precisely, we prove
P
j
Theorem 1.1. Let p(z) = z m n−m
j=0 aj z be a polynomial of degree n, having all its
zeros in |z| ≤ k, k ≤ 1, with a zero of order m at z = 0. Then for β with |β| < k n−m
and s ≥ 1
Z
2π
(1.6)
0
s 1s
0
0 iθ
mm
i(m−1)θ
p (e ) +
dθ
β̄e
n
k
≥ n − (n −
m)Cs(k)
vol. 10, iss. 1, art. 23, 2009
Z
2π
0
s 1s
0
iθ
m
imθ
p(e ) +
dθ ,
β̄e
kn
where m0 = min |p(z)|,
|z|=k
Cs(k) =
1
2π
Z
2π
|Sc + eiθ |s dθ
− 1s
and
0
an−m−1 an−m + 1
.
Sc =
an−m−1 1
k 2 + n−m an−m 1
n−m
Corollary 1.2. If p(z) is a polynomial of degree n, having all its zeros in |z| ≤ 1,
with a zero of order m at z = 0, then for s ≥ 1
2π
iθ
s
|p(e )| dθ
(1.7)
0
1s
≥ n − (n −
m)Cs(1)
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By taking k = 1 and β = 0 in Theorem 1.1, we obtain:
Z
Integral Mean Estimates
K.K. Dewan, N. Singh,
B. Chanam and A. Mir
Z
2π
iθ
s
|p(e )| dθ
0
1s
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where
Cs(1) = 1
1 R 2π
|1 + eiθ |s dθ
2π 0
1s .
By letting s → ∞ in Theorem 1.1, we obtain
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Corollary 1.3. Let p(z) = z m n−m
j=0 aj z be a polynomial of degree n, having all its
zeros in |z| ≤ k, k ≤ 1, with a zero of order m at z = 0. Then for β with |β| < k n−m
Integral Mean Estimates
K.K. Dewan, N. Singh,
B. Chanam and A. Mir
vol. 10, iss. 1, art. 23, 2009
0
0
0
m
+
nS
m
mm
c
max p (z) + n β̄z m−1 ≥
max p(z) + n β̄z m ,
|z|=1
|z|=1
k
1 + Sc
k
(1.8)
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0
where m and Sc are as defined in Theorem 1.1.
By choosing the argument of β suitably and letting |β| → k n−m in Corollary 1.3,
we obtain the following result.
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Corollary 1.4. Let p(z) = z m n−m
j=0 aj z be a polynomial of degree n, having all
its zeros in |z| ≤ k, k ≤ 1, with a zero of order m at z = 0. Then
m + nSc
(n − m)Sc m0
0
(1.9)
max |p (z)| ≥
max |p(z)| +
,
|z|=1
|z|=1
1 + Sc
1 + Sc k m
where m0 and Sc are as defined in Theorem 1.1.
Let Dα p(z) denote the polar derivative of the polynomial p(z) of degree n with
respect to the point α. Then
Dα p(z) = np(z) + (α − z)p0 (z) .
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The polynomial Dα p(z) is of degree at most (n − 1) and it generalizes the ordinary
derivative in the sense that
Dα p(z)
= p0 (z) .
α→∞
α
lim
(1.10)
Our next result generalizes as well as improving upon the inequality (1.4), which
in turns, gives a generalization as well as improvements of inequalities (1.3), (1.2)
and (1.1) in terms of the polar derivatives of Lp inequalities.
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Theorem 1.5. If p(z) = an z n + nj=µ an−j z n−j , 1 ≤ µ ≤ n, is a polynomial of
degree n, having all its zeros in |z| ≤ k, k ≤ 1, then for every real or complex
numbers α and β with |α| ≥ k µ and |β| ≤ 1 and for each r > 0
Integral Mean Estimates
K.K. Dewan, N. Singh,
B. Chanam and A. Mir
vol. 10, iss. 1, art. 23, 2009
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(1.11) max |Dα p(z)|
Contents
|z|=1
≥
R
n(|α| − k µ )
2π
0
|1 + k µ eiθ |r dθ
Z
1
r
0
2π
r r1
0
iθ
n
βm
i(n−1)θ p(e ) +
+ n−µ m0 ,
e
dθ
n−µ
k
k
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0
where m = min |p(z)|.
|z|=k
Dividing both sides of (1.11) by |α|, letting |α| → ∞ and noting that (1.10), we
obtain
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Corollary 1.6. If p(z) = an z n + nj=µ an−j z n−j , 1 ≤ µ ≤ n, is a polynomial of
degree n, having all its zeros in |z| ≤ k, k ≤ 1, then for every real or complex
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number β with |β| ≤ 1, for each r > 0
(1.12)
max |p0 (z)|
|z|=1
≥
R 2π
0
Z
n
|1 +
k µ eiθ |r dθ
r1
0
2π
r r1
0
iθ
βm
i(n−1)θ
dθ ,
p(e ) +
e
k n−µ
where m0 = min |p(z)|.
|z|=k
Integral Mean Estimates
K.K. Dewan, N. Singh,
B. Chanam and A. Mir
vol. 10, iss. 1, art. 23, 2009
Remark 1. Letting r → ∞ in (1.12) and
Pchoosing the argument of β suitably with
|β| = 1, it follows that, if p(z) = an z n + nj=µ an−j z n−j , 1 ≤ µ ≤ n, is a polynomial
of degree n, having all its zeros in |z| ≤ k, k ≤ 1, then
n
1
0
(1.13)
max |p (z)| ≥
max |p(z)| + n−µ min |p(z)| .
|z|=k
|z|=1
(1 + k µ ) |z|=1
k
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Inequality (1.13) was already proved by Aziz and Shah [2].
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2.
Lemmas
For the proofs of these theorems we need the following lemmas.
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Lemma 2.1. Let p(z) = nj=0 aj z j be a polynomial of degree n having no zeros in
|z| < k, k ≥ 1. Then for s ≥ 1
Z
2π
0
iθ
s
1s
|p (e )| dθ
(2.1)
Z
≤ nSs
2π
iθ
s
|p(e )| dθ
0
1s
,
0
vol. 10, iss. 1, art. 23, 2009
where
Ss =
1
2π
Z
2π
|Sc0 + eiθ |s dθ
1s
h i
1 a1 +
1
n a0
Sc0 =
.
1 a1 2
1 + n a0 k
k2
and
0
The above lemma is due to Dewan, Bhat and Pukhta [3].
The following lemma is due to Rather [7].
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Lemma 2.2. Let p(z) = an z n + nj=µ an−j z n−j , 1 ≤ µ ≤ n, be a polynomial of
degree n having all its zero in |z| ≤ k, k ≤ 1. Then
(2.2)
Integral Mean Estimates
K.K. Dewan, N. Singh,
B. Chanam and A. Mir
k µ |p0 (z)| ≥ |q 0 (z)| +
n
k n−µ
min |p(z)| for |z| = 1,
|z|=k
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where q(z) = z n p
1
z̄
.
3.
Proofs of The Theorems
Proof of Theorem 1.1. Let
p(z) = z
m
n−m
X
aj z j = z m φ(z),
(say)
j=0
where φ(z) is a polynomial of degree n − m, with the property that
φ(0) 6= 0 .
Integral Mean Estimates
K.K. Dewan, N. Singh,
B. Chanam and A. Mir
vol. 10, iss. 1, art. 23, 2009
Then
1
1
n−m
q(z) = z p
=z
φ
z̄
z̄
n
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is also a polynomial of degree n − m and has no zeros in |z| < k1 , k1 ≥ 1. Now if
m0
1 1
n
m0 = min1 |q(z)| = min1 z p
= n min |p(z)| = n ,
z̄ k |z|=k
k
|z|= k
|z|= k JJ
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then, by Rouche’s theorem, the polynomial
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q(z) + m0 βz n−m ,
Contents
|β| < k n−m ,
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of degree n − m, will also have no zeros in |z| <
≥ 1. Hence, by Lemma 2.1,
we have for s ≥ 1 and |β| < k n−m
s 1s
Z 2π 0
0 iθ
q (e ) + m βei(n−m−1)θ (n − m) dθ
n
k
0
s 1s
Z 2π 0
iθ
m
(k)
i(n−m)θ
dθ ,
q(e ) +
≤ (n − m)Cs
βe
kn
0
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which implies
2π
Z
(3.1)
0
s 1s
0
m
imθ
iθ
iθ
0
iθ
np(e ) − e p (e ) + β̄ (n − m)e dθ
kn
s 1s
Z 2π 0
m
imθ p(eiθ ) +
≤ (n − m)Cs(k)
β̄e
dθ .
kn
0
Now by Minkowski’s inequality, we have for s ≥ 1 and |β| < k n−m
Z
2π
n
0
s 1s
0
iθ
p(e ) + m β̄eimθ dθ
n
k
s 1s
Z 2π 0
m
imθ
iθ
0
iθ
iθ
np(e ) +
dθ
≤
β̄(n
−
m)e
−
e
p
(e
)
kn
0
s 1s
Z 2π 0
iθ 0 iθ
mm
imθ e p (e ) +
+
β̄e
dθ ,
kn
0
which implies, by using inequality (3.1)
Z
n
0
2π
s 1s
0
iθ
m
imθ
p(e ) +
dθ
β̄e
kn
Z
(k)
≤ (n − m)Cs
s 1s
2π 0
p(eiθ ) + m β̄eimθ dθ
n
k
0
s 1s
Z 2π 0
0 iθ
m
i(m−1)θ
dθ ,
p (e ) + m β̄e
+
kn
0
and the Theorem 1.1 follows.
Integral Mean Estimates
K.K. Dewan, N. Singh,
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vol. 10, iss. 1, art. 23, 2009
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Proof of Theorem 1.5. Since q(z) = z n p z̄1 so that p(z) = z n q
have
1
1
0
n−1
n−2 0
(3.2)
p (z) = nz q
−z q
,
z̄
z̄
1
z̄
, therefore, we
which implies
(3.3)
|p0 (z)| = |nq(z) − zq 0 (z)| for |z| = 1 .
Using (3.2) in (2.2), we get for 1 ≤ µ ≤ n
|q 0 (z)| +
vol. 10, iss. 1, art. 23, 2009
0
mn
≤ k µ |nq(z) − zq 0 (z)| for |z| = 1 .
k n−µ
Now, from the above inequality, for every complex β with |β| ≤ 1, we get, for
|z| = 1
0 0
0
m
n
q (z) + β̄
≤ |q 0 (z)| + m n
k n−µ k n−µ
(3.4)
≤ k µ |nq(z) − zq 0 (z)| .
For every real or complex number α with |α| ≥ k µ , we have
|Dα p(z)| = |np(z) + (α − z)p0 (z)|
≥ |α| |p0 (z)| − |np(z) − zp0 (z)|,
which gives by interchanging the roles of p(z) and q(z) in (3.3) for |z| = 1
|Dα p(z)| ≥ |α||p0 (z)| − |q 0 (z)|
(3.5)
Integral Mean Estimates
K.K. Dewan, N. Singh,
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≥ |α||p0 (z)| − k µ |p0 (z)| +
m0 n
k n−µ
(using (2.2)).
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Since p(z) has all its zeros in |z| ≤ k ≤ 1, by the Gauss-Lucas theorem, all the zeros
of p0 (z) also lie in |z| ≤ 1. This implies that the polynomial
1
n−1 0
z p
= nq(z) − zq 0 (z)
z̄
has all its zeros in |z| ≥
1
k
≥ 1. Therefore, it follows from (3.4) that the function
m0 n
zq (z) + β̄ n−µ z
k
w(z) = µ
k (nq(z) − zq 0 (z))
0
(3.6)
is analytic for |z| ≤ 1 and |w(z)| ≤ 1 for |z| ≤ 1. Furthermore w(0) = 0. Thus the
function 1 + k µ w(z) is a subordinate to the function 1 + k µ z in |z| ≤ 1. Hence by a
well-known property of subordination [4], we have for r > 0 and for 0 ≤ θ < 2π,
Z 2π
Z 2π
µ
iθ r
(3.7)
|1 + k w(e )| dθ ≤
|1 + k µ eiθ |r dθ .
0
0
Also from (3.6), we have
m0 n
nq(z) + β̄ n−µ z
k
1 + k µ w(z) =
,
nq(z) − zq 0 (z)
or
0
nq(z) + β̄ m n z = |1 + k µ w(z)||p0 (z)| for |z| = 1,
n−µ
k
which implies
0
m
(3.8)
n p(z) + β n−µ z n−1 = |1 + k µ w(z)||p0 (z)| for |z| = 1.
k
Integral Mean Estimates
K.K. Dewan, N. Singh,
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Now combining (3.7) and (3.8), we get
r
Z 2π Z 2π
0
iθ
m
r
i(n−1)θ
n
|1 + k µ eiθ |r |p0 (eiθ )|r dθ .
p(e ) + β k n−µ e
dθ ≤
0
0
Using (3.5) in the above inequality, we obtain
r
Z 2π iθ
m0 i(n−1)θ r
µ r
n (|α| − k )
dθ
p(e ) + β k n−µ e
0
r
Z 2π
nm0
µ iθ r
≤
|1 + k e | dθ max |Dα p(z)| − n−µ ,
|z|=1
k
0
from which we obtain the required result.
Integral Mean Estimates
K.K. Dewan, N. Singh,
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vol. 10, iss. 1, art. 23, 2009
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References
[1] A. AZIZ, Integral mean estimate for polynomials with restricted zeros, J. Approx. Theory, 55 (1988), 232–239.
[2] A. AZIZ AND W.M. SHAH, An integral mean estimate for polynomials, Indian
J. Pure Appl. Math., 28(10) (1997), 1413–1419.
[3] K.K. DEWAN, A. BHAT AND M.S. PUKHTA, Inequalities concerning the Lp
norm of a polynomial, J. Math. Anal. Appl., 224 (1998), 14–21.
[4] E. HILLE, Analytic Function Theory, Vol. II, Ginn and Company, New York,
Toronto, 1962.
[5] M.A. MALIK, On the derivative of a polynomial, J. London Math. Soc., 1
(1969), 57–60.
[6] M.A. MALIK, An integral mean estimate for polynomials, Proc. Amer. Math.
Soc., 91 (1984), 281–284.
[7] N.A. RATHER, Extremal properties and location of the zeros of polynomials,
Ph.D. Thesis submitted to the University of Kashmir, 1998.
[8] W. RUDIN, Real and Complex Analysis, Tata McGraw-Hill Pub. Co., 1977.
[9] A.E. TAYLOR, Introduction to Functional Analysis, John Wiley and Sons Inc.,
New York, 1958.
[10] P. TURÁN, Über die Ableitung von Polynomen, Compositio Math., 7 (1939),
89–95.
[11] A. ZYGMUND, A remark on conjugate series, Proc. London Math. Soc., 34(2)
(1932), 392–400.
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vol. 10, iss. 1, art. 23, 2009
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