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Inequalities Concerning theLp-Norm of a Polynomial
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
224, 14]21 Ž1998.
AY985980
Inequalities Concerning the L p-Norm of a Polynomial K. K. Dewan Department of Mathematics, Faculty of Natural Sciences, Jamia Millia Islamia, New Delhi- 110025, India
Aijaz Ahmad Bhat Department of Mathematics, Go¨ ernment Degree College, Sopore, Kashmir- 193201, India
and Mohammad Sayeed Pukhta Department of Mathematics, Go¨ ernment Degree College, Baramulla, Kashmir- 193201, India Submitted by A. M. Fink Received November 26, 1997
In this paper we obtain L p , p G 1, inequalities for the class of polynomials having no zeros in < z < - K, K G 1. Our result generalizes as well as improves upon some well known results. Q 1998 Academic Press Key Words: Polynomials; derivative; zeros; inequalities; extremal problems.
1. INTRODUCTION AND STATEMENT OF RESULTS Let pŽ z . s ݨns0 a¨ z ¨ be a polynomial of degree n. Then we have
ž
2p
H0
p9 Ž e i u .
q
1rq
du
/
Fn
ž
2p
H0
14 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
p Ž e iu .
q
1rq
du
/
,
q G 1, Ž 1.1.
THE L
p
15
NORM OF A POLYNOMIAL
and
ž
2p
H0
q
p Ž Re i u .
1rq
du
F Rn
/
ž
2p
1rq
q
p Ž e iu .
H0
du
/
q G 1, R ) 1.
,
Ž 1.2. Inequality Ž1.1. is due to Zygmund w12x and inequality Ž1.2. is easy to prove. For pŽ z . / 0 in < z < - 1 the inequalities Ž1.1. and Ž1.2. have been replaced, respectively, by 2p
žH
p9 Ž e i u .
0
q
1rq
du
F n Ž Cq .
/
1rq
2p
p Ž e iu .
žH
q
0
1rq
du
/
q G 1,
,
Ž 1.3. where C q s 2yq 'p G Ž 12 q q 1 . G Ž 12 q q
1 2
.,
and 2p
žH
p Ž Re i u .
q
0
1rq
du
F Ž Kq.
/
1rq
2p
žH
p Ž e iu .
0
q
1rq
du
/
,
R G 1, q G 1,
Ž 1.4.
where Kq s
2p
1 q R ne i nu
H0
q
du
2p
H0
1 q e i n u du .
Inequality Ž1.3. was first proved by de-Bruijn w2x Žfor another proof, see Rahman w11x., and Ž1.4. is due to Boas and Rahman w1x. The extremal polynomial in each case is pŽ z . s a q b z n, < a < s < b <. Dewan and Govil w4x Žsee also Govil and Jain w7x. considered the class of polynomials pŽ z . satisfying pŽ z . ' z n pŽ1rz . and obtained a sharp inequality analogous to Ž1.1.. There is greater interest attached to the case when pŽ z . does not vanish in the circle < z < - K, where K is a positive number. The answer to this more general question for the case when K F 1 and q s 2 was given by Rahman w10x. For the case when K G 1, the following result is known w5x. THEOREM A. If pŽ z . s ݨns0 a¨ z ¨ is a polynomial of degree n, ha¨ ing no zeroes in < z < - K, K G 1, then, for q G 1, we ha¨ e 2p
žH
0
p9 Ž e i u .
q
1rq
du
/
F n Ž Eq .
2p
žH
0
p Ž e iu .
q
1rq
du
/
,
Ž 1.5.
16
DEWAN, BHAT, AND PUKHTA
where 2p
Eq s 2 p
ž
K q e iu
H0
q
1rq
du
/
.
Theorem A is not sharp and the sharp inequality does not seem to be obtainable even for q s 2. In this direction Dewan and Bidkham w3x obtained an inequality for q s 2, the bound of which is, in general, better than the bound obtained by Ž1.5.. In this paper we shall generalize Theorem A as well as improve upon the bound obtained in inequality Ž1.5. by involving the coefficients < a0 < and < am <, 1 F m F n. Besides this, we also prove an inequality analogous to Ž1.2.. We prove THEOREM 1. If pŽ z . s a0 q ݨnsm a¨ z ¨ is a polynomial of degree n such that pŽ z . has no zeros in the disk < z < - K, K G 1, then, for q G 1,
ž
2p
H0
q
p9 Ž e i u .
1rq
du
F nS q
/
ž
2p
p Ž e iu .
H0
1rq
q
du
/
,
Ž 1.6.
where 2p
S q s 2p
ž
H0
Sm c q e i u
q
1rq
du
/
and Sm c s
K mq1 Ž mrn amra0 K m y1 q 1 . 1 q mrn amra0 K m q1
.
The result is best possible in the case K s 1 and equality holds for the polynomial pŽ z . s 1 q z n. For K s m s 1, Theorem 1 reduces to Ž1.3. due to de-Bruijn w2x. For m s 1, Theorem 1 is, in general, an improvement over Theorem A due to Govil and Rahman w5x. To show this we have to prove that for q G 1, S q F Eq , which is equivalent to
ž
2p
2p
H0
Sm c q e i u
q
1rq
du
F 2p
/ ž
2p
H0
K q e iu
q
1rq
du
/
or 2p
H0
K q e iu
q
du F
2p
H0
Sm c q e i u
q
du .
Ž 1.7.
THE L
p
17
NORM OF A POLYNOMIAL
Now to prove Ž1.7. it is sufficient to prove that K F Sm c or K 2 Ž 1rn a1ra0 q 1 .
KF
1 q 1rn a1ra0 K 2
for m s 1,
which on simplification gives
ž
K2
1 a1 n a0
Ž K y 1. F K Ž K y 1. ,
/
which implies a1
F
a0
n K
.
Ž 1.8.
Since Ž1.8. is always true Žsee w6, pp. 320]321x., hence Ž1.7. follows. If we assume m s 2 in Theorem 1, then we get the following: COROLLARY 1. If pŽ z . s ݨns0 a¨ z ¨ is a polynomial of degree n, ha¨ ing no zeros in < z < - K, K G 1, and p9Ž0. s 0, then, for q G 1,
ž
2p
p9 Ž e i u .
H0
F n 2p
ž
q
1rq
du 2p
H0
/ K qe 2
iu
q
1rq
du
/ ž
2p
p Ž e iu .
H0
q
1rq
du
/
. Ž 1.9.
If is easy to see that Corollary 1 also provides a generalization and improvement to Theorem A due to Govil and Rahman w5x and to a result proved by Dewan and Bidkham w3x. For q s 1, Theorem 1 yields COROLLARY 2. 2p
H0
Let pŽ z . be the same as in Theorem 1. Then
p9 Ž e1 u . du F n 2p
ž
2p
H0
Sm c q e i u du
/
2p
H0
p Ž e i u . du . Ž 1.10.
THEOREM 2. If pŽ z . s a0 q ݨnsm a¨ z ¨ is a polynomial of degree n, ha¨ ing no zeroes in < z < - K, K G 1, then, for R ) 1, 2p
H0
p Ž Re i u . du F S1 Ž R n y 1 . q 1 4
2p
H0
p Ž e i u . du ,
Ž 1.11.
where S1 s 2prH02 p < Sm c q e i u < du and Sm c is the same as defined in Theorem 1.
18
DEWAN, BHAT, AND PUKHTA
2. A LEMMA We need the following result due to Qazi Žsee w8, p. 339x. for the proof of the theorems. LEMMA. If pŽ z . s a0 q ݨnsm a¨ z ¨ has no zeroes in < z < - K, K G 1, then, for < z < s 1, < p9 Ž z . < F
1 q mrn amra0 K mq1
mrn amra0 K
my1
1
q1 K
< q9 Ž z . < ,
m q1
Ž 2.1.
where q Ž z . s z n p Ž 1rz . 4 . Proof of Theorem 1. By a known result due to de-Bruijn w2, p. 1271x, we have for every q G 1 and real a , 2p
H0
e i u p9 Ž e i u .
iu
pŽ e . y
qe
n
iŽ a q u .
q
p9 Ž e i u .
du F
n
2p
H0
p Ž e iu .
q
du .
Integrating both the sides of the above inequality with respect to a from 0 to 2p , we get 2p
H0
2p
da
pŽ e . y
H0
F 2p
e i u p9 Ž e i u .
iu
2p
n
p Ž e iu .
H0
q
du ,
qe
iŽ a q u .
p9 Ž e i u .
q
du
n
q G 1.
Ž 3.1.
Note that p9Ž e i u . can be zero only at a countable number of points. Besides, we can clearly invert the order of integration on the left-hand side of Ž3.1.. Therefore, 2p
H0
da
2p
H0
s
s
2p
H0
2p
H0
iu
pŽ e . y da
2p
H0
p9 Ž e i u . n
e i u p9 Ž e i u . n
qe
q
p9 Ž e i u .
ia
e q
n q
du
2p
H0
ia
e q
iŽ a q u .
p9 Ž e i u .
q
n
du
np Ž e i u . y e i u p9 Ž e i u .
q
du
e i u p9 Ž e i u . np Ž e i u . y e i u p9 Ž e i u . e i u p9 Ž e i u .
q
d a . Ž 3.2.
p
THE L
19
NORM OF A POLYNOMIAL
Thus for 0 F u - 2p and every q G 1 and using the lemma, we get 2p
np Ž e i u . y e i u p9 Ž e i u .
ia
e q
H0
s
e q
2p
G
H0
da
e i u p9 Ž e i u . q
q9 Ž e i u .
eia q
2p
q
np Ž e i u . y e i u p9 Ž e i u .
ia
H0 H0
da
e i u p9 Ž e i u .
2p
s
q
da
p9 Ž e i u . q
e i a q Sm c
da .
Ž 3.3.
Combining inequalities Ž3.1., Ž3.2., and Ž3.3., we get, for q G 1, 2p
H0
p9 Ž e i u . n
q
2p
du
H0
e i a q Sm c
q
2p
d a F 2p
p Ž e iu .
H0
q
du ,
which gives 2p
H0
p9 Ž e i u .
q
2p
du F 2 p n q
ž
H0
e i a q Sm c
q
2p
da
/H
p Ž e iu .
0
q
du
from which the theorem follows. Remark. The proof of the theorem can also be obtained from the arguments used in Rahman w9x for proving de-Bruijn’s theorem. Proof of Theorem 2. For each 0 F u - 2p , we have p Ž Re i u . y p Ž e i u . s
R iu
H1
e p9 Ž re i u . dr ,
R ) 1,
which implies p Ž Re i u . F
R
H1
p9 Ž re i u . dr q p Ž e i u . ,
R ) 1.
Integrating both sides of the above inequality with respect to u from 0 to 2p , we get 2p
H0
p Ž Re i u . du F
R
2p
H1 drH0
p9 Ž re i u . du q
2p
H0
p Ž e i u . du . Ž 3.4.
20
DEWAN, BHAT, AND PUKHTA
Now since the polynomial p9Ž z . is of degree Ž n y 1., therefore, inequality Ž1.2., for q s 1 reduces to 2p
H0
2p
p9 Ž Re i u . du F R ny1
H0
p9 Ž e i u . du .
Applying the above inequality to Ž3.4., we get 2p
H0
p Ž Re i u . du F
R
H1
2p
r ny1 dr
H0
p9 Ž e i u . du q
2p
H0
p Ž e i u . du ,
which on using Corollary 2 gives 2p
H0
p Ž Re i u . du F nS1
2p
H0
p Ž e i u . du
R
H1
r ny1 dr q
2p
H0
p Ž e i u . du ,
Ž 3.5. where S1 s 2p
2p
H0
Sm c q e i u du
and Sm c s
K mq1 Ž mrn amra0 K m y1 q 1 . 1 q mrn amra0 K m q1
Thus inequality Ž3.5. is equivalent to 2p
H0
p Ž Re i u . du F S1 Ž R n y 1 . q 1 4
2p
H0
p Ž e i u . du ,
which completes the proof of the theorem.
REFERENCES 1. R. P. Boas, Jr. and Q. I. Rahman, L p inequalities for polynomials and entire functions, Arch. Rational Mech. Analy. 11 Ž1962., 34]39. 2. N. G. de-Bruijn, Inequalities concerning polynomials in the complex domain, Nederl. Akad. Wetensch Proc. Ser. A 21 Ž1914., 14]22. 3. K. K. Dewan and M. Bidkham, Some integral inequalities for polynomials, Glas. Mat. Ser. III 28 Ž1993., 227]234. 4. K. K. Dewan and N. K. Govil, An inequality for self-inversive polynomials, J. Math. Anal. Appl. 95 Ž1982., 490.
THE L
p
NORM OF A POLYNOMIAL
21
5. N. K. Govil and Q. I. Rahman, Functions of exponential type not vanishing in a half plane and related polynomials, Trans. Amer. Math. Soc. 137 Ž1969., 501]517. 6. N. K. Govil, Q. I. Rahman, and G. Schmeisser, On the derivative of a polynomial, Illinois J. Math. 23 Ž1979., 319]330. 7. N. K. Govil and V. K. Jain, An integral inequality for entire functions of exponential type, Ann. Uni¨ . Mariae Curie-Skłodowska Sect. A 39 Ž1985., 57]60. 8. M. A. Qazi, On the maximum modulus of polynomials, Proc. Amer. Math. Soc. 115 Ž1992., 337]343. 9. Q. I. Rahman, Inequalities concerning polynomials and trigonometric polynomials, J. Math. Anal. Appl. 6 Ž1963., 303]324. 10. Q. I. Rahman, Some inequalities for polynomials and related entire functions II, Canad. Math. Bull. 7 Ž1964., 573]594. 11. Q. I. Rahman, Functions of exponential type, Trans. Amer. Math. Soc. 135 Ž1969., 295]305. 12. A. Zygmund, A remark on conjugate series, Proc. London Math. Soc. 34 Ž1932., 392]400.
224, 14]21 Ž1998.
AY985980
Inequalities Concerning the L p-Norm of a Polynomial K. K. Dewan Department of Mathematics, Faculty of Natural Sciences, Jamia Millia Islamia, New Delhi- 110025, India
Aijaz Ahmad Bhat Department of Mathematics, Go¨ ernment Degree College, Sopore, Kashmir- 193201, India
and Mohammad Sayeed Pukhta Department of Mathematics, Go¨ ernment Degree College, Baramulla, Kashmir- 193201, India Submitted by A. M. Fink Received November 26, 1997
In this paper we obtain L p , p G 1, inequalities for the class of polynomials having no zeros in < z < - K, K G 1. Our result generalizes as well as improves upon some well known results. Q 1998 Academic Press Key Words: Polynomials; derivative; zeros; inequalities; extremal problems.
1. INTRODUCTION AND STATEMENT OF RESULTS Let pŽ z . s ݨns0 a¨ z ¨ be a polynomial of degree n. Then we have
ž
2p
H0
p9 Ž e i u .
q
1rq
du
/
Fn
ž
2p
H0
14 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
p Ž e iu .
q
1rq
du
/
,
q G 1, Ž 1.1.
THE L
p
15
NORM OF A POLYNOMIAL
and
ž
2p
H0
q
p Ž Re i u .
1rq
du
F Rn
/
ž
2p
1rq
q
p Ž e iu .
H0
du
/
q G 1, R ) 1.
,
Ž 1.2. Inequality Ž1.1. is due to Zygmund w12x and inequality Ž1.2. is easy to prove. For pŽ z . / 0 in < z < - 1 the inequalities Ž1.1. and Ž1.2. have been replaced, respectively, by 2p
žH
p9 Ž e i u .
0
q
1rq
du
F n Ž Cq .
/
1rq
2p
p Ž e iu .
žH
q
0
1rq
du
/
q G 1,
,
Ž 1.3. where C q s 2yq 'p G Ž 12 q q 1 . G Ž 12 q q
1 2
.,
and 2p
žH
p Ž Re i u .
q
0
1rq
du
F Ž Kq.
/
1rq
2p
žH
p Ž e iu .
0
q
1rq
du
/
,
R G 1, q G 1,
Ž 1.4.
where Kq s
2p
1 q R ne i nu
H0
q
du
2p
H0
1 q e i n u du .
Inequality Ž1.3. was first proved by de-Bruijn w2x Žfor another proof, see Rahman w11x., and Ž1.4. is due to Boas and Rahman w1x. The extremal polynomial in each case is pŽ z . s a q b z n, < a < s < b <. Dewan and Govil w4x Žsee also Govil and Jain w7x. considered the class of polynomials pŽ z . satisfying pŽ z . ' z n pŽ1rz . and obtained a sharp inequality analogous to Ž1.1.. There is greater interest attached to the case when pŽ z . does not vanish in the circle < z < - K, where K is a positive number. The answer to this more general question for the case when K F 1 and q s 2 was given by Rahman w10x. For the case when K G 1, the following result is known w5x. THEOREM A. If pŽ z . s ݨns0 a¨ z ¨ is a polynomial of degree n, ha¨ ing no zeroes in < z < - K, K G 1, then, for q G 1, we ha¨ e 2p
žH
0
p9 Ž e i u .
q
1rq
du
/
F n Ž Eq .
2p
žH
0
p Ž e iu .
q
1rq
du
/
,
Ž 1.5.
16
DEWAN, BHAT, AND PUKHTA
where 2p
Eq s 2 p
ž
K q e iu
H0
q
1rq
du
/
.
Theorem A is not sharp and the sharp inequality does not seem to be obtainable even for q s 2. In this direction Dewan and Bidkham w3x obtained an inequality for q s 2, the bound of which is, in general, better than the bound obtained by Ž1.5.. In this paper we shall generalize Theorem A as well as improve upon the bound obtained in inequality Ž1.5. by involving the coefficients < a0 < and < am <, 1 F m F n. Besides this, we also prove an inequality analogous to Ž1.2.. We prove THEOREM 1. If pŽ z . s a0 q ݨnsm a¨ z ¨ is a polynomial of degree n such that pŽ z . has no zeros in the disk < z < - K, K G 1, then, for q G 1,
ž
2p
H0
q
p9 Ž e i u .
1rq
du
F nS q
/
ž
2p
p Ž e iu .
H0
1rq
q
du
/
,
Ž 1.6.
where 2p
S q s 2p
ž
H0
Sm c q e i u
q
1rq
du
/
and Sm c s
K mq1 Ž mrn amra0 K m y1 q 1 . 1 q mrn amra0 K m q1
.
The result is best possible in the case K s 1 and equality holds for the polynomial pŽ z . s 1 q z n. For K s m s 1, Theorem 1 reduces to Ž1.3. due to de-Bruijn w2x. For m s 1, Theorem 1 is, in general, an improvement over Theorem A due to Govil and Rahman w5x. To show this we have to prove that for q G 1, S q F Eq , which is equivalent to
ž
2p
2p
H0
Sm c q e i u
q
1rq
du
F 2p
/ ž
2p
H0
K q e iu
q
1rq
du
/
or 2p
H0
K q e iu
q
du F
2p
H0
Sm c q e i u
q
du .
Ž 1.7.
THE L
p
17
NORM OF A POLYNOMIAL
Now to prove Ž1.7. it is sufficient to prove that K F Sm c or K 2 Ž 1rn a1ra0 q 1 .
KF
1 q 1rn a1ra0 K 2
for m s 1,
which on simplification gives
ž
K2
1 a1 n a0
Ž K y 1. F K Ž K y 1. ,
/
which implies a1
F
a0
n K
.
Ž 1.8.
Since Ž1.8. is always true Žsee w6, pp. 320]321x., hence Ž1.7. follows. If we assume m s 2 in Theorem 1, then we get the following: COROLLARY 1. If pŽ z . s ݨns0 a¨ z ¨ is a polynomial of degree n, ha¨ ing no zeros in < z < - K, K G 1, and p9Ž0. s 0, then, for q G 1,
ž
2p
p9 Ž e i u .
H0
F n 2p
ž
q
1rq
du 2p
H0
/ K qe 2
iu
q
1rq
du
/ ž
2p
p Ž e iu .
H0
q
1rq
du
/
. Ž 1.9.
If is easy to see that Corollary 1 also provides a generalization and improvement to Theorem A due to Govil and Rahman w5x and to a result proved by Dewan and Bidkham w3x. For q s 1, Theorem 1 yields COROLLARY 2. 2p
H0
Let pŽ z . be the same as in Theorem 1. Then
p9 Ž e1 u . du F n 2p
ž
2p
H0
Sm c q e i u du
/
2p
H0
p Ž e i u . du . Ž 1.10.
THEOREM 2. If pŽ z . s a0 q ݨnsm a¨ z ¨ is a polynomial of degree n, ha¨ ing no zeroes in < z < - K, K G 1, then, for R ) 1, 2p
H0
p Ž Re i u . du F S1 Ž R n y 1 . q 1 4
2p
H0
p Ž e i u . du ,
Ž 1.11.
where S1 s 2prH02 p < Sm c q e i u < du and Sm c is the same as defined in Theorem 1.
18
DEWAN, BHAT, AND PUKHTA
2. A LEMMA We need the following result due to Qazi Žsee w8, p. 339x. for the proof of the theorems. LEMMA. If pŽ z . s a0 q ݨnsm a¨ z ¨ has no zeroes in < z < - K, K G 1, then, for < z < s 1, < p9 Ž z . < F
1 q mrn amra0 K mq1
mrn amra0 K
my1
1
q1 K
< q9 Ž z . < ,
m q1
Ž 2.1.
where q Ž z . s z n p Ž 1rz . 4 . Proof of Theorem 1. By a known result due to de-Bruijn w2, p. 1271x, we have for every q G 1 and real a , 2p
H0
e i u p9 Ž e i u .
iu
pŽ e . y
qe
n
iŽ a q u .
q
p9 Ž e i u .
du F
n
2p
H0
p Ž e iu .
q
du .
Integrating both the sides of the above inequality with respect to a from 0 to 2p , we get 2p
H0
2p
da
pŽ e . y
H0
F 2p
e i u p9 Ž e i u .
iu
2p
n
p Ž e iu .
H0
q
du ,
qe
iŽ a q u .
p9 Ž e i u .
q
du
n
q G 1.
Ž 3.1.
Note that p9Ž e i u . can be zero only at a countable number of points. Besides, we can clearly invert the order of integration on the left-hand side of Ž3.1.. Therefore, 2p
H0
da
2p
H0
s
s
2p
H0
2p
H0
iu
pŽ e . y da
2p
H0
p9 Ž e i u . n
e i u p9 Ž e i u . n
qe
q
p9 Ž e i u .
ia
e q
n q
du
2p
H0
ia
e q
iŽ a q u .
p9 Ž e i u .
q
n
du
np Ž e i u . y e i u p9 Ž e i u .
q
du
e i u p9 Ž e i u . np Ž e i u . y e i u p9 Ž e i u . e i u p9 Ž e i u .
q
d a . Ž 3.2.
p
THE L
19
NORM OF A POLYNOMIAL
Thus for 0 F u - 2p and every q G 1 and using the lemma, we get 2p
np Ž e i u . y e i u p9 Ž e i u .
ia
e q
H0
s
e q
2p
G
H0
da
e i u p9 Ž e i u . q
q9 Ž e i u .
eia q
2p
q
np Ž e i u . y e i u p9 Ž e i u .
ia
H0 H0
da
e i u p9 Ž e i u .
2p
s
q
da
p9 Ž e i u . q
e i a q Sm c
da .
Ž 3.3.
Combining inequalities Ž3.1., Ž3.2., and Ž3.3., we get, for q G 1, 2p
H0
p9 Ž e i u . n
q
2p
du
H0
e i a q Sm c
q
2p
d a F 2p
p Ž e iu .
H0
q
du ,
which gives 2p
H0
p9 Ž e i u .
q
2p
du F 2 p n q
ž
H0
e i a q Sm c
q
2p
da
/H
p Ž e iu .
0
q
du
from which the theorem follows. Remark. The proof of the theorem can also be obtained from the arguments used in Rahman w9x for proving de-Bruijn’s theorem. Proof of Theorem 2. For each 0 F u - 2p , we have p Ž Re i u . y p Ž e i u . s
R iu
H1
e p9 Ž re i u . dr ,
R ) 1,
which implies p Ž Re i u . F
R
H1
p9 Ž re i u . dr q p Ž e i u . ,
R ) 1.
Integrating both sides of the above inequality with respect to u from 0 to 2p , we get 2p
H0
p Ž Re i u . du F
R
2p
H1 drH0
p9 Ž re i u . du q
2p
H0
p Ž e i u . du . Ž 3.4.
20
DEWAN, BHAT, AND PUKHTA
Now since the polynomial p9Ž z . is of degree Ž n y 1., therefore, inequality Ž1.2., for q s 1 reduces to 2p
H0
2p
p9 Ž Re i u . du F R ny1
H0
p9 Ž e i u . du .
Applying the above inequality to Ž3.4., we get 2p
H0
p Ž Re i u . du F
R
H1
2p
r ny1 dr
H0
p9 Ž e i u . du q
2p
H0
p Ž e i u . du ,
which on using Corollary 2 gives 2p
H0
p Ž Re i u . du F nS1
2p
H0
p Ž e i u . du
R
H1
r ny1 dr q
2p
H0
p Ž e i u . du ,
Ž 3.5. where S1 s 2p
2p
H0
Sm c q e i u du
and Sm c s
K mq1 Ž mrn amra0 K m y1 q 1 . 1 q mrn amra0 K m q1
Thus inequality Ž3.5. is equivalent to 2p
H0
p Ž Re i u . du F S1 Ž R n y 1 . q 1 4
2p
H0
p Ž e i u . du ,
which completes the proof of the theorem.
REFERENCES 1. R. P. Boas, Jr. and Q. I. Rahman, L p inequalities for polynomials and entire functions, Arch. Rational Mech. Analy. 11 Ž1962., 34]39. 2. N. G. de-Bruijn, Inequalities concerning polynomials in the complex domain, Nederl. Akad. Wetensch Proc. Ser. A 21 Ž1914., 14]22. 3. K. K. Dewan and M. Bidkham, Some integral inequalities for polynomials, Glas. Mat. Ser. III 28 Ž1993., 227]234. 4. K. K. Dewan and N. K. Govil, An inequality for self-inversive polynomials, J. Math. Anal. Appl. 95 Ž1982., 490.
THE L
p
NORM OF A POLYNOMIAL
21
5. N. K. Govil and Q. I. Rahman, Functions of exponential type not vanishing in a half plane and related polynomials, Trans. Amer. Math. Soc. 137 Ž1969., 501]517. 6. N. K. Govil, Q. I. Rahman, and G. Schmeisser, On the derivative of a polynomial, Illinois J. Math. 23 Ž1979., 319]330. 7. N. K. Govil and V. K. Jain, An integral inequality for entire functions of exponential type, Ann. Uni¨ . Mariae Curie-Skłodowska Sect. A 39 Ž1985., 57]60. 8. M. A. Qazi, On the maximum modulus of polynomials, Proc. Amer. Math. Soc. 115 Ž1992., 337]343. 9. Q. I. Rahman, Inequalities concerning polynomials and trigonometric polynomials, J. Math. Anal. Appl. 6 Ž1963., 303]324. 10. Q. I. Rahman, Some inequalities for polynomials and related entire functions II, Canad. Math. Bull. 7 Ž1964., 573]594. 11. Q. I. Rahman, Functions of exponential type, Trans. Amer. Math. Soc. 135 Ž1969., 295]305. 12. A. Zygmund, A remark on conjugate series, Proc. London Math. Soc. 34 Ž1932., 392]400.