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An integral inequality for cosine polynomials
Applied Mathematics and Computation 249 (2014) 532–534
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
An integral inequality for cosine polynomials Horst Alzer a, Allal Guessab b,⇑ a b
Morsbacher Str. 10, 51545 Waldbröl, Germany Laboratoire de Mathématiques et de leurs Applications, UMR CNRS 4152, Université de Pau et des Pays de l’Adour, 64000 Pau, France
a r t i c l e
i n f o
a b s t r a c t Let
Keywords: Integral inequality Cosine polynomial Convex function
T n ðxÞ ¼
n X 1 a0 þ ak cosðkxÞ: 2 k¼1
We prove that if ak ðk ¼ 0; 1; . . . ; nÞ is an increasing sequence of real numbers, then for any function f which is increasing and convex on the real line we have
1
p
! Z p n jT n ðxÞj 1 X f ak ; dx P f Ln n þ 1 k¼0 0
where
Ln ¼
1
p
Z p n 1 X cosðkxÞdx þ 2 0 k¼1
denotes the Lebesgue constant. This extends and refines a result due to Fejes (1939). Ó 2014 Elsevier Inc. All rights reserved.
In what follows, we denote by T n a cosine polynomial of degree n with real coefficients,
T n ðxÞ ¼
n X 1 a0 þ ak cosðkxÞ: 2 k¼1
In 1938, Sidon [5] published the following inequality. Proposition 1. Let ak ðk ¼ 0; 1; . . . ; nÞ be non-negative real numbers. Then there exists a constant C such that
Z p
jT n ðxÞjdx > C log n min ak : 06k6n
0
One year later, Fejes [1] proved a refinement of this result (see also [3, Section 2.2.2]). Proposition 2. Let ak ðk ¼ 0; 1; . . . ; nÞ be non-negative real numbers. Then,
1
Z p
p
0
jT n ðxÞjdx P Ln min ak ; 06k6n
⇑ Corresponding author. E-mail addresses: [email protected] (H. Alzer), [email protected] (A. Guessab). http://dx.doi.org/10.1016/j.amc.2014.10.086 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
ð1Þ
H. Alzer, A. Guessab / Applied Mathematics and Computation 249 (2014) 532–534
533
where Ln denotes the Lebesgue constant
Ln ¼
1
p
Z p n 1 X cosðkxÞdx: þ 2 0 k¼1
Setting a0 ¼ a1 ¼ ¼ an ¼ 1 reveals that Ln is the best possible constant in (1). Two questions arise: (i) Is it possible to improve the lower bound in (1)? Rp (ii) Does there exist a positive lower bound for 0 jT n ðxÞjdx, if some of the coefficients are negative? P It is the aim of this note to show that in (1) we can replace the factor min06k6n ak by the arithmetic mean nk¼0 ak =ðn þ 1Þ under the assumption that the coefficients are increasing. This leads to an improvement of (1). Moreover, if a0 6 a1 6 6 an Rp P and nk¼0 ak > 0, then we obtain a positive lower bound for 0 jT n ðxÞjdx. Actually, we prove a bit more. We present a lower Rp bound for 0 f ðjT n ðxÞj=Ln Þdx, where f is an increasing and convex function. In order to prove our result we need two classical inequalities. Tchebychef’s inequality. Let xk and yk ðk ¼ 0; 1; . . . ; nÞ be real numbers with
x 0 6 x1 6 6 xn
and y0 6 y1 6 6 yn :
Then, n n n X X X xk yk 6 ðn þ 1Þ xk yk : k¼0
k¼0
ð2Þ
k¼0
The sign of equality holds in (2) if and only if x0 ¼ ¼ xn or y0 ¼ ¼ yn . Jensen’s inequality. Let g be convex on ðc; dÞ and let h be integrable on ½a; b with c < hðtÞ < d. Then,
1 g ba
Z
!
b
hðtÞdt
6
a
1 ba
Z
b
g ðhðtÞÞdt:
a
Proofs can be found, for instance, in [4, Section 2.5] and [2, Section 6.14], respectively. Our general inequality can be stated as follows. Theorem. Let ak ðk ¼ 0; 1; . . . ; nÞ be real numbers satisfying
a0 6 a1 6 6 an :
ð3Þ
Then, for any function f which is increasing and convex on the real line we have
1
p
! Z p n jT n ðxÞj 1 X dx P f f ak : Ln n þ 1 k¼0 0
ð4Þ
If a0 < an and f is strictly increasing, then (4) holds with ‘‘>’’ instead of ‘‘P’’. Proof. We follow the method of proof given in [1]. As shown in [1] we have
Ln ¼
n X 1 pn0 þ pnk ; 2 k¼1
with
pnk ¼
1
Z p
p
0
ð5Þ
cosðkxÞ sgn sin ððn þ 1=2ÞxÞdx ¼
1 8ð2n þ 1Þ X
p2
1
2 2 m¼0 ð2n þ 1Þ ð2m þ 1Þ 4k
2
:
ð6Þ
Applying jAj ¼ A sgn A P A sgn B we obtain
1
Z p
p
0
jT n ðxÞjdx P
1
Z p
p
0
T n ðxÞ sgn sin ððn þ 1=2ÞxÞdx ¼
n X 1 pn0 a0 þ pnk ak : 2 k¼1
ð7Þ
From the series representation in (6) we find
1 p < pn1 < < pnn : 2 n0
ð8Þ
Using (3) and (8) we conclude from Tchebychef’s inequality that
! n n n X X X 1 1 1 pn0 a0 þ pnk ak P pn0 þ pnk ak ; 2 nþ1 2 k¼1 k¼1 k¼0
ð9Þ
534
H. Alzer, A. Guessab / Applied Mathematics and Computation 249 (2014) 532–534
with equality if and only if a0 ¼ an . Next, we combine (7), (9) and (5). This gives
1
p
Z p n jT n ðxÞj 1 X dx P ak : Ln n þ 1 k¼0 0
ð10Þ
Since f is increasing, we get
f
1
p
! Z p n jT n ðxÞj 1 X dx P f ak : Ln n þ 1 k¼0 0
ð11Þ
An application of Jensen’s inequality leads to
1
p
Z p Z p jT n ðxÞj 1 jT n ðxÞj dx P f f dx : Ln Ln p 0 0
Combining the last two inequalities yields (4). Moreover, if a0 < an and f is strictly increasing, then the inequalities (9)–(11) are strict, so that we obtain (4) with ‘‘>’’ instead of ‘‘P’’. h Remark 1. If f is decreasing and concave, then the converse of (4) holds. 2. The special case f ðxÞ ¼ x reveals that if a0 6 a1 6 6 an , then the following improvement of (1) is valid:
1
Z p
p
0
jT n ðxÞjdx P Ln
n 1 X ak P Ln min ak : 06k6n n þ 1 k¼0
References [1] [2] [3] [4] [5]
L. Fejes, Two inequalities concerning trigonometric polynomials, J. London Math. Soc. 14 (1939) 44–46. G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Camb. Univ. Press, Cambridge, 1952. G.V. Milovanovic´, D.S. Mitrinovic´, Th.M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Sci, Singapore, 1994. D.S. Mitrinovic´, Analytic Inequalities, Springer, New York, 1970. S. Sidon, Über Fourier-Koeffizienten, J. London Math. Soc. 13 (1938) 181–183.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
An integral inequality for cosine polynomials Horst Alzer a, Allal Guessab b,⇑ a b
Morsbacher Str. 10, 51545 Waldbröl, Germany Laboratoire de Mathématiques et de leurs Applications, UMR CNRS 4152, Université de Pau et des Pays de l’Adour, 64000 Pau, France
a r t i c l e
i n f o
a b s t r a c t Let
Keywords: Integral inequality Cosine polynomial Convex function
T n ðxÞ ¼
n X 1 a0 þ ak cosðkxÞ: 2 k¼1
We prove that if ak ðk ¼ 0; 1; . . . ; nÞ is an increasing sequence of real numbers, then for any function f which is increasing and convex on the real line we have
1
p
! Z p n jT n ðxÞj 1 X f ak ; dx P f Ln n þ 1 k¼0 0
where
Ln ¼
1
p
Z p n 1 X cosðkxÞdx þ 2 0 k¼1
denotes the Lebesgue constant. This extends and refines a result due to Fejes (1939). Ó 2014 Elsevier Inc. All rights reserved.
In what follows, we denote by T n a cosine polynomial of degree n with real coefficients,
T n ðxÞ ¼
n X 1 a0 þ ak cosðkxÞ: 2 k¼1
In 1938, Sidon [5] published the following inequality. Proposition 1. Let ak ðk ¼ 0; 1; . . . ; nÞ be non-negative real numbers. Then there exists a constant C such that
Z p
jT n ðxÞjdx > C log n min ak : 06k6n
0
One year later, Fejes [1] proved a refinement of this result (see also [3, Section 2.2.2]). Proposition 2. Let ak ðk ¼ 0; 1; . . . ; nÞ be non-negative real numbers. Then,
1
Z p
p
0
jT n ðxÞjdx P Ln min ak ; 06k6n
⇑ Corresponding author. E-mail addresses: [email protected] (H. Alzer), [email protected] (A. Guessab). http://dx.doi.org/10.1016/j.amc.2014.10.086 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
ð1Þ
H. Alzer, A. Guessab / Applied Mathematics and Computation 249 (2014) 532–534
533
where Ln denotes the Lebesgue constant
Ln ¼
1
p
Z p n 1 X cosðkxÞdx: þ 2 0 k¼1
Setting a0 ¼ a1 ¼ ¼ an ¼ 1 reveals that Ln is the best possible constant in (1). Two questions arise: (i) Is it possible to improve the lower bound in (1)? Rp (ii) Does there exist a positive lower bound for 0 jT n ðxÞjdx, if some of the coefficients are negative? P It is the aim of this note to show that in (1) we can replace the factor min06k6n ak by the arithmetic mean nk¼0 ak =ðn þ 1Þ under the assumption that the coefficients are increasing. This leads to an improvement of (1). Moreover, if a0 6 a1 6 6 an Rp P and nk¼0 ak > 0, then we obtain a positive lower bound for 0 jT n ðxÞjdx. Actually, we prove a bit more. We present a lower Rp bound for 0 f ðjT n ðxÞj=Ln Þdx, where f is an increasing and convex function. In order to prove our result we need two classical inequalities. Tchebychef’s inequality. Let xk and yk ðk ¼ 0; 1; . . . ; nÞ be real numbers with
x 0 6 x1 6 6 xn
and y0 6 y1 6 6 yn :
Then, n n n X X X xk yk 6 ðn þ 1Þ xk yk : k¼0
k¼0
ð2Þ
k¼0
The sign of equality holds in (2) if and only if x0 ¼ ¼ xn or y0 ¼ ¼ yn . Jensen’s inequality. Let g be convex on ðc; dÞ and let h be integrable on ½a; b with c < hðtÞ < d. Then,
1 g ba
Z
!
b
hðtÞdt
6
a
1 ba
Z
b
g ðhðtÞÞdt:
a
Proofs can be found, for instance, in [4, Section 2.5] and [2, Section 6.14], respectively. Our general inequality can be stated as follows. Theorem. Let ak ðk ¼ 0; 1; . . . ; nÞ be real numbers satisfying
a0 6 a1 6 6 an :
ð3Þ
Then, for any function f which is increasing and convex on the real line we have
1
p
! Z p n jT n ðxÞj 1 X dx P f f ak : Ln n þ 1 k¼0 0
ð4Þ
If a0 < an and f is strictly increasing, then (4) holds with ‘‘>’’ instead of ‘‘P’’. Proof. We follow the method of proof given in [1]. As shown in [1] we have
Ln ¼
n X 1 pn0 þ pnk ; 2 k¼1
with
pnk ¼
1
Z p
p
0
ð5Þ
cosðkxÞ sgn sin ððn þ 1=2ÞxÞdx ¼
1 8ð2n þ 1Þ X
p2
1
2 2 m¼0 ð2n þ 1Þ ð2m þ 1Þ 4k
2
:
ð6Þ
Applying jAj ¼ A sgn A P A sgn B we obtain
1
Z p
p
0
jT n ðxÞjdx P
1
Z p
p
0
T n ðxÞ sgn sin ððn þ 1=2ÞxÞdx ¼
n X 1 pn0 a0 þ pnk ak : 2 k¼1
ð7Þ
From the series representation in (6) we find
1 p < pn1 < < pnn : 2 n0
ð8Þ
Using (3) and (8) we conclude from Tchebychef’s inequality that
! n n n X X X 1 1 1 pn0 a0 þ pnk ak P pn0 þ pnk ak ; 2 nþ1 2 k¼1 k¼1 k¼0
ð9Þ
534
H. Alzer, A. Guessab / Applied Mathematics and Computation 249 (2014) 532–534
with equality if and only if a0 ¼ an . Next, we combine (7), (9) and (5). This gives
1
p
Z p n jT n ðxÞj 1 X dx P ak : Ln n þ 1 k¼0 0
ð10Þ
Since f is increasing, we get
f
1
p
! Z p n jT n ðxÞj 1 X dx P f ak : Ln n þ 1 k¼0 0
ð11Þ
An application of Jensen’s inequality leads to
1
p
Z p Z p jT n ðxÞj 1 jT n ðxÞj dx P f f dx : Ln Ln p 0 0
Combining the last two inequalities yields (4). Moreover, if a0 < an and f is strictly increasing, then the inequalities (9)–(11) are strict, so that we obtain (4) with ‘‘>’’ instead of ‘‘P’’. h Remark 1. If f is decreasing and concave, then the converse of (4) holds. 2. The special case f ðxÞ ¼ x reveals that if a0 6 a1 6 6 an , then the following improvement of (1) is valid:
1
Z p
p
0
jT n ðxÞjdx P Ln
n 1 X ak P Ln min ak : 06k6n n þ 1 k¼0
References [1] [2] [3] [4] [5]
L. Fejes, Two inequalities concerning trigonometric polynomials, J. London Math. Soc. 14 (1939) 44–46. G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Camb. Univ. Press, Cambridge, 1952. G.V. Milovanovic´, D.S. Mitrinovic´, Th.M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Sci, Singapore, 1994. D.S. Mitrinovic´, Analytic Inequalities, Springer, New York, 1970. S. Sidon, Über Fourier-Koeffizienten, J. London Math. Soc. 13 (1938) 181–183.