J. of Inequal. & Appl., 1997, Vol. 1, pp. 301-310
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Discrete Inequalities of Wirtinger’s
Type for Higher Differences
GRADIMIR V.
MILOVANOVI and IGOR :. MILOVANOVI_,
Faculty of Electronic Engineering, Department of Mathematics,
P. O. Box 73, 18001 Ni, Yugoslavia
e-mail: grade @ gauss.elfak.ni.ac, yu
(Received 28 July 1996)
Discrete version of Wirtinger’s type inequality for higher differences,
An,m
x
k=
where lm
[m/2], Um
n
2< (Amxk)2 nn,m
x
k=
k=Im
[m/2] and
mmxk
(--1)
Xk+m_i
i=0
is considered. Under some conditions, the best constants
An,m and Bn,m e deteined.
Keywords: Discrete inequality; difference of higher order; eigenvalue; eigenvector.
AMS 1991 Subject classifications: Primary 26D 15; Secondary 41 A44.
1 INTRODUCTION AND PRELIMINARIES
In 1 (see also [2]) we presented a general method for finding the best possible
constants
An and Bn in inequalities of the form
An
px <_
k=l
rlc(xlk=0
<
px,
Bn
(1.1)
k=l
This work was supported in part by the Serbian Scientific Foundation, grant number 04M03.
301
G.V. MILOVANOVI( and I.. MILOVANOVI(
302
where p (pk) and r (rk) are given weight sequences and e (x) is an
arbitrary sequence of the real numbers. The basic discrete inequalities of the
1 were given by K. Fan, O. Taussky, and J. Todd
form (1.1) for p r
[3]. Here, we mention some references in this direction [4-8].
The first results for the second difference were proved by Fan, Taussky
and Todd [3]:
THEOREM 1.1
numbers, then
-
If XO(-" 0),
n-1
(Xk
2Xk+l Xk+2) 2 >__
k=0
THEOREI 1.2
XO Xl, Xn+
If XO, Xl
Xn, Xn+l are
n
7
16 sin4
A sin
with equality in (1.2) if and only if x
A is an arbitrary constant.
("- O) are given real
Xn, Xn+l
XI,X2
2(n + 1)
kTr
(1.2)
k=l
n, where
1 2,
k
n+l
x:,2
given real numbers such that
Xn and
n
x
(1.3)
0,
k=l
then
(Xk
2Xk+l + Xk+2 >_
16 sin4
k=0
nn
x:.
(1.4)
k=l
The equality in (1.4) is attained if and only if
xk
A cos
(2k
2n
k-- 1,2
n,
where A is an arbitrary constant.
A converse inequality of (1.2) was proved by Lunter [9], Yin [10] and
Chen [11] (see also Agarwal [8]).
/f X0(= 0),
Xl,X2
Xn, xn+l
n-1
(x
2x+ + x+e) e _<
16cos4
k=0
with equality in (1.5) if and only if x:
Chen [11 also proved the following result:
n
zr
2(n + 1)
A(-1) sin
where A is an arbitrary constant.
(= 0) are given real
kr
n-I-
1
i" k
2
xc’
1 2
(1.5)
n,
DISCRETE INEQUALITIES OF WIRTINGER’S TYPE
THEOREM 1.4 /f
xo x and Xn+
Xn, Xn/
X0, Xl
303
are given real numbers such that
then
Xn,
n-1
n
2X+l -F x+2) 2 <_ 16cos4
(x
k=O
n
x,
k=l
with equality holding if and only if
x
A(-1) sin
(2k- 1)re
1,2,... ,n,
k
where A is an arbitrary constant.
In this case, the n
n symmetric matrix corresponding to the quadratic
form
n-1
F2
__,(Xk
2Xk+l + Xk+2) 2
(Hn,2, ,)
k=O
2 -3
1
1
6 -4
-3
6 -4
1 -4
1
"’. "’. "’. "’. "’.
6 -4
1 -4
1
6 -3
1 -4
2_
1 -3
n matrix
This matrix is the square of the n
1 -1
1
2 -1
-1
2 -1
-1
(1.6)
2 -1
-1
1
The eigenvalues of Hn are
.v
v(Hn)
4cos 2
(n
v
+ 1)zr
2n
v=l
and therefore, the largest eigenvalue of Hn is
)n(Hn)
4cos 2
nn > ,kn-1 (Hn).
n,
G.V. MILOVANOVI( and I.;. MILOVANOVI0
304
The corresponding eigenvector is a n
(-1) sin
Xn
(2v
1
v--- 1,2
2n
where
Xnn
Xln X2n
n.
Thus, the largest eigenvalue of Hn,2 is
n(Hn,2)-- 16cos4
nn > ,n-l(Hn,2),
and the associated eigenvector is x n
Notice that the minimal eigenvalue of the matrix Hn (and also Hn,2) is
0. Therefore, the condition (1.3) must be included in Theorem 1.2 (see
X
Agarwal [8, Ch. 11]) and the best constant is the square of the relevant
eigenvalue
re
(n- 1)re
4 sin 2
2 4 COS 2 2n
2n
For any n-dimensional vector a
[xl x2
Xn ]r, Pfeffer [12]
introduced a periodically extended n-vector by setting Xi+rn
xi for
n and r 6 N, and used the rnth difference of a given by
1, 2
(m)
Amxn ]T, where
Amxl Amx2
m
(--1) m-r
Amxi
r---O
()
r
l<i<n,
Xi-[m/2]+r,
in order to prove the following result:
THEOREM 1.5
If a is a periodically extended n-vector and (1.3) holds, then
(X(rn) x(rn)) >
with equality case
(4sin
2
if and only if a is the periodic extension of a vector of the
form Clu + Czv, where
U
Ul
Un
U2
T
and
vk
sin
v
Vl
132
k
1
Vn ]T
have the following components
uk
cos
2krc
,
g/
2krc
,
n
and C1 and C2 are arbitrary real constants.
n,
DISCRETE INEQUALITIES OF WIRTINGER’S TYPE
305
2 MAIN RESULTS
In this paper we consider inequalities of the form
xk2
An,m
k=
where Im
1
(Amxk) 2 < Bn,m
<
[m/2] and
n
(--1)
AmXk
(2.1)
k=
k--Im
[m/2], Um
2
xk,
Xk+m-i.
i=0
/m
(AmXk) 2 for m
The quadratic form Fm
x
Vl
n-I
n-1
2xkxk+l
2x + xn2
-Jr-
1 reduces to
k=l
k=2
with corresponding tridiagonal symmetric matrix Itn
Hn, given by (1.6).
We consider inequalities (2.1) under conditions
Xs
Xl-s,
Xn+l-s
and define
Xn+s
(lm <
S
< 0)
(2.2)
AJxl-[j/2]
AJ x2-[j/2]
x (j)
(2.3)
AJXn-[j/2]
The quadratic form
Fm
can be expressed then in the following form
Um
Fm_ Fro(x,)_
(Amxk) 2 ((m), x,(m)),
where
(0)
Xl
x2
Xn
At the begining we prove three auxiliary results:
(2.4)
G.V. MILOVANOVI( and I.. MILOVANOVI(
306
LEMMA 2.1 If j is an even integer, under conditions (2.2), we have that
AJ+lx-[j/2]
Proof Let q
0 or q
A J+ Xq-[j/2]
A2p+lxq-p
zxJ+lxn-[j/2]
and
0
O.
(2.5)
2p we have
n. Putting j
2p+l
+1
E (-1)i (2p
\
,
i=0
Xq+p+l-i
2p+l
,(_i
x//_i +
i=0
P
E(--1) (2p+l) Xq+p+l-i E(--1)
i--0
2p + 1
x//_
Xq-p+i
i=0
E(_I)i
i=0
t
(-
i=p+l
( +)
2p
1
(Xq+p+ l_i
Xq-p+i
0
because of the conditions (2.2).
LEMMA 2.2 If j is an even integer, under conditions (2.2), we have that
_x(j+2),
Hnx(j)
where the matrix Hn is given by (1.6).
Proof We have
AJ xI_[j/2]
A j x2-[j/2]
--AJ+2xI_[j/2
Un x (J)
(2.6)
AJ+2xn
2 -[j/2]
A J Xn- 1-[j/2] "1- A J Xn-[j/2]
Since
AJ+2x_[j/2]
AJ+lxl_[j/2] AJ+lx_[j/2]
and
AJ+2xn_l_[j/2]
AJ+lxn_[j/2] AJ+lxn_l_[j/2],
DISCRETE INEQUALITIES OF WIRTINGER’S TYPE
307
because of Lemma 2.1, we conclude that
AJ+2x_[j/2]- AJ+IxI_[j/2]
and
AJ+2xn_l_[j/2] -"--AJ+lxn_l_[j/2],
respectively. Therefore,
AJ xI-[j/2]
AJ x2-[j/2]
AJ+lxl
[j/2]
_AJ +2x-[j/2]
and
--AJ Xn-l-[j/2] + Aj xn-[j/2]
AJ+I Xn-l-[j/2]
AJ+2xn -1-[j/2].
Then (2.6) becomes
AJ+2xI-[(j+2)/2]
AJ+2x2-[(j+2)/2]
AJ+2x_[j/2
AJ+2xl_[j/2]
_x(J+2).
Hnx(J)
A j +2x n_ 1-[(j +2)/21
A]+2xn-2-[j/2]
AJ+2Xn-l-[j/2]
AJ+2xn-[(j+2)/2]
LEMMA 2.3 If j is an even integer, under conditions (2.2), we have that
(x(j), x(j+2))
_((j+l), (j+l)).
Proof Let j is an even integer. Using (2.3) we have
n
(X
(j)
X
(j+2))
E Ajxk-[j/2] AJ+2xk
-1-[j/21
k=l
E AJxk-[J/z](AJxk-l-[j/2]
E AJxk-[j/2](AJxk+l-[j/2]
2AJxk-[j/2] + AJXk+l-[j/2])
k=l
n
AJxk-[j/2])
k=l
}2
k=l
AJ Xk-[j/2]AJ+lxk-[j/2]
k=l
n
E Ajxk-[j/2lAj+lxk-[j/2l
k=l
E Aj Xk-[j/2]AJ+l
E AJxk+l-[j/2lAJ+lxk-[j/2]"
Xk-l-[j/2]
k=l
n-1
k=O
G.V. MILOVANOVI( and I.. MILOVANOVI(
308
Because of (2.5) we can write
(X (j), x(J+2))--
-AJxk-[j/2]AJ+" 1Xk_[j/2] AJxk+I_[j/2]A
j+l
Xk-[j/2]
k=l
k=l
n
2
1Xk-[j
k=l
Since j is an even integer we have that
(Aj + lxk-[(j+l)/2] )2
(x(j) x(j+2))
_(x(J+I), x(J+l)).
k--1
Now, we give the main result:
THEOREM 2.4
If Xl, X2
Xn are given real numbers and conditions (2.2)
are satisfied, then
n
bl
(AmXk)
2
<_ 4m COS 2m
k=lm
where lm 1
if and only if
[m/2] and Um
A(-1) sin
Xk
(2.7)
Xk’
k=l
n
(2k
[m/2]. The equality in (2.7) is attained
1)re
k-- 1,2
n,
where A is an arbitrary constant.
Proof We will prove that the corresponding matrix of the quadratic form
(2.4) is exactly the mth power of the matrix Hn
Hn,1 so that the best
constant in the right inequality (2.1), i.e., (2.7), is given by
Bn,m
Evidently,
An,m
4m cs2m
7g
2n
O.
Let m be an even integer. Then, using Lemma 2.2, we find
Fm
(x,(m), x, (m)) (Un3(m-2), Unx,(m-2)),
DISCRETE INEQUALITIES OF WIRTINGER’S TYPE
309
<>, Hnm/’ <>)
em
Similarly, for an odd m, using Lemmas 2.3 and 2.4, we obtain
Frn
(x (m), x(m)) _((m--1), (m+l)) ((m-1), nn(m-1)).
Now, using Lemma 2.2 again, we find
(nn(m-1)/2x (0), nn(m+l)/2 (0)) (anm, ).
Frn
By restriction (1.3), we can obtain the following result:
TI-IEORM 2.5 If xl, X2
Xn are given real numbers and conditions (2.2)
and (1.3) are satisfied, then
4m sinZm
where lm 1
if and only if
7z"
2--
[m/2] and Um
xk
Acos
2
[m /2]. The equality in (2.8) is attained
1)Tr
(2k
2n
(2.8)
k--Im
k=l
n
(Amxl) 2
<
k
1,2
n,
where A is an arbitrary constant.
For other generalizations of discrete Wirtinger’s inequalities see 13-15].
There are also generalizations for multidimensional sequences and partial
differences (see 16] and 17]).
Acknowledgements
This work was supported in part by the Serbian Scientific Foundation, grant
number 04M03.
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