Journal of Inequalities in Pure and
Applied Mathematics
ON CERTAIN INEQUALITIES RELATED TO THE
SEITZ INEQUALITY
volume 5, issue 2, article 39,
2004.
LIANG-CHENG WANG AND JIA-GUI LUO
Department of Mathematics
Daxian Teacher’s College
Dazhou 635000, Sichuan Province
The People’s Republic of China.
EMail: wangliangcheng@163.com
Department of Mathematics
Zhongshan University
Guangzhou 510275
People’s Republic of China.
Received 18 August, 2003;
accepted 11 April, 2004.
Communicated by: F. Qi
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
In this paper, we investigate the monotonicity of difference results from the G.
Seitz inequality. An application is given, with some resulting inequalities.
2000 Mathematics Subject Classification: 26D15
Key words: G. Seitz inequality, Convex function, Difference, Monotonicity, Exponential convex.
The first author is partially supported by the Key Research Foundation of the Daxian
Teacher’s College under Grant 2003–81.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
References
On Certain Inequalities Related
to the Seitz Inequality
Liang-Cheng Wang and Jia-Gui
Luo
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1.
Introduction
For a given positive integer n ≥ 2, let X = (x1 , x2 , . . . , xn ), Y = (y1 , y2 , . . . , yn ),
U = (u1 , u2 , . . . , un ) and Z = (z1 , z2 , . . . , zn )P
be known sequences of real
numbers, and let ti > 0 (i = 1, 2, . . . , n), Tj = ji=1 ti (j = 1, 2, . . . , n) and
aij (i, j = 1, 2, . . . , n) be known real numbers. Define the functions A, J, C,
W and G by
4
A(n) =
n
X
n
Y
xi − n
i=1
! n1
xi
On Certain Inequalities Related
to the Seitz Inequality
(xi > 0, i = 1, 2, . . . , n),
i=1
4
J(n) =
n
X
ti f (vi ) − Tn f
i=1
1
Tn
n
X
Liang-Cheng Wang and Jia-Gui
Luo
!
ti vi ,
i=1
Title Page
where f is convex function on the interval I and vi ∈ I(i = 1, 2, . . . , n),
"
4
C(n) =
n
X
!
x2i
n
X
i=1
4
W (n) = Tn
n
X
!# 12
yi2
−
i=1
ti xi zi −
n
X
i=1
n
X
Contents
JJ
J
xi yi ,
i=1
!
ti xi
i=1
n
X
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!
ti zi ,
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i=1
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and
4
G(n) =
n
X
i,j=1
!
aij xi zj
n
X
i,j=1
!
aij yi uj
−
n
X
i,j=1
!
aij xi uj
n
X
i,j=1
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!
aij yi zj
.
J. Ineq. Pure and Appl. Math. 5(2) Art. 39, 2004
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Rade investigated the monotonicity of difference for A − G mean inequality,
and obtained the following inequality [2]
A(n) ≥ A(n − 1).
(1.1)
P. M. Vasić and J. E. Pečarić generalized inequality (1.1) to convex functions,
and obtained the following inequality [4, 6]
J(n) ≥ J(n − 1).
(1.2)
On Certain Inequalities Related
to the Seitz Inequality
Recently the first author and Xu Zhang studied inequality (1.2) in depth, and
obtained some inequalities. L.-C. Wang also obtained some applications, one
of them is the following inequality [8]
Liang-Cheng Wang and Jia-Gui
Luo
C(n) ≥ C(n − 1).
Title Page
(1.3)
Contents
Inequality (1.3) resulted from the Cauchy inequality
(1.4)
n
X
i=1
!
x2i
n
X
i=1
!
yi2
≥
n
X
!2
xi yi
.
i=1
In [7], L.-C. Wang proved the following inequality
(1.5)
W (n) ≥ W (n − 1),
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with X and Z both increasing or both decreasing. If one of X or Z is increasing
and the other decreasing, then the inequality (1.5) reverses.
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Inequality (1.5) resulted from the following Chebyshev inequality
! n
!
n
n
X
X
X
(1.6)
Tn
ti xi zi ≥
ti xi
ti zi ,
i=1
i=1
i=1
with X and Z both increasing or both decreasing. If one of X or Z is increasing
and the other decreasing, then the inequality (1.6) reverses.
Assume that i, j, r, s ∈ N such that 1 ≤ i < j ≤ n and 1 ≤ r < s ≤ n, we
have
xi xj zr zs (1.7)
y i y j ur us ≥ 0
On Certain Inequalities Related
to the Seitz Inequality
Liang-Cheng Wang and Jia-Gui
Luo
and
air ais
ajr ajs
(1.8)
≥ 0.
Title Page
Contents
When both (1.7) and (1.8) are true, the following inequality by G. Seitz [1]
holds:
n
n
P
P
aij xi zj
aij yi zj
i,j=1
i,j=1
(1.9)
≥ P
.
n
n
P
aij xi uj
aij yi uj
i,j=1
i,j=1
(1.10) X = Z,
Y =U
and aij =
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If
(
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J
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1 i=j
0 i 6= j
(i, j = 1, 2, . . . , n),
J. Ineq. Pure and Appl. Math. 5(2) Art. 39, 2004
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then inequality (1.9) changes into (1.4). If
(
(1.11) Y = U = (1, 1, . . . , 1) and aij =
ti i = j
0 i 6= j
(i, j = 1, 2, . . . , n),
then inequality (1.9) changes into (1.6).
In this paper, we investigate inequality (1.9) in depth, obtaining the following
main result.
Theorem 1.1. If both inequalities (1.7) and (1.8) are true, then we have
On Certain Inequalities Related
to the Seitz Inequality
G(n) ≥ G(n − 1).
Liang-Cheng Wang and Jia-Gui
Luo
Remark 1.1. If we put (1.10) and (1.11) into (1.12), then (1.12) becomes (1.3)
and (1.5), respectively. Hence, (1.12) is an extension of (1.3) and (1.5).
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(1.12)
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2.
Proof of Theorem 1.1
Using
aij xi zj ain yi un = aij yi zj ain xi un
aij yi uj
(i, j = 1, 2, . . . , n − 1),
ain xi zn = aij xi uj ain yi zn
(i, j = 1, 2, . . . , n − 1),
and (1.7) – (1.8), we have
(2.1)
n−1
X
aij xi zj
i,j=1
n−1
X
ain yi un −
i=1
+
n−1
X
n−1 X
n−1
X
aij yi uj
aij xi zj
n−1
X
n−1
X
n−1
X
aij xi uj
i,j=1
akn yk un −
aij yi uj
i=1 j=1
=
ain xi zn −
n−1
X
k=1
Liang-Cheng Wang and Jia-Gui
Luo
akn xk zn −
n−1
X
ain yi zn
Title Page
i=1
n−1 X
n−1
X
aij yi zj
i=1 j=1
k=1
n−1
n−1 X
X
n−1 X
n−1
X
n−1
X
ain xi un
i=1
i=1
i=1 j=1
+
aij yi zj
n−1
X
i,j=1
i,j=1
=
n−1
X
On Certain Inequalities Related
to the Seitz Inequality
n−1
n−1 X
X
i=1 j=1
n−1
X
Contents
akn xk un
k=1
aij xi uj
n−1
X
akn yk zn
k=1
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aij akn
× xi zj yk un + xk zn yi uj − yi zj xk un − xi uj yk zn
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i=1 j=1 k=1,k6=i
JJ
J
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=
n−1
n−2 X
n−1
X
X
j=1
+
i=1 k=2,i<k
!
n−1 X
n−2
X
aij akn
i=2 k=1,i>k
× xi zj yk un + xk zn yi uj − yi zj xk un − xi uj yk zn
=
n−1 X
n−2 X
n−1
X
aij akn xi zj yk un + xk zn yi uj − yi zj xk un − xi uj yk zn
j=1 i=1 k=2,i<k
+
n−1 X
n−2
n−1 X
X
akj ain xk zj yi un + xi zn yk uj − yk zj xi un − xk uj yi zn
On Certain Inequalities Related
to the Seitz Inequality
j=1 k=2 i=1,k>i
=
n−1
X
X aij akn − akj ain
Liang-Cheng Wang and Jia-Gui
Luo
j=1 1≤i<k<n
× xi zj yk un + xk zn yi uj − yi zj xk un − xi uj yk zn
n−1
X
X aij ain xi xk zj zn =
akj akn yi yk uj un ≥ 0.
j=1 1≤i<k<n
Using
aij xi zj anj yn uj = aij xi uj anj yn zj
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(i, j = 1, 2, . . . , n − 1),
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aij yi uj
anj xn zj = aij yi zj
anj xn uj
(i, j = 1, 2, . . . , n − 1),
Page 8 of 19
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(1.7) – (1.8) and the same method as in the proof of (2.1), we obtain
(2.2)
n−1
X
aij xi zj
i,j=1
n−1
X
anj yn uj −
j=1
+
n−1
X
aij xi uj
n−1
X
i,j=1
aij yi uj
i,j=1
=
n−1
X
n−1
X
anj xn zj −
j=1
n−1 X
n−1 X
n−1
X
anj yn zj
j=1
n−1
X
aij yi zj
i,j=1
n−1
X
anj xn uj
j=1
aij ank
On Certain Inequalities Related
to the Seitz Inequality
i=1 j=1 k=1,k6=j
=
× xi zj yn uk + yi uj xn zk − xi uj yn zk − yi zj xn uk
X aij aik xi xn zj zk anj ank yi yn uj uk ≥ 0.
n−1
X
Liang-Cheng Wang and Jia-Gui
Luo
Title Page
i=1 1≤j<k<n
Contents
Using
ain yi un ajn xj zn = ain yi zn ajn xj un
and
(i, j = 1, 2, . . . , n)
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ani yn ui
anj xn zj = ani yn zi
anj xn uj
(i, j = 1, 2, . . . , n − 1),
Close
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we obtain
(2.3)
n
X
i=1
ain yi un
n
X
j=1
ajn xj zn −
n
X
i=1
ain yi zn
n
X
j=1
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ajn xj un = 0
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and
n−1
X
(2.4)
ani yn ui
i=1
n−1
X
anj xn zj −
j=1
n−1
X
ani yn zi
i=1
n−1
X
respectively.
Using
ann yn un anj xn zj = anj yn zj ann xn un
anj yn uj
anj xn uj = 0,
j=1
ann xn zn = ann yn zn anj xn uj
(j = 1, 2, . . . , n − 1),
On Certain Inequalities Related
to the Seitz Inequality
(j = 1, 2, . . . , n − 1),
Liang-Cheng Wang and Jia-Gui
Luo
and (1.7) – (1.8), we have
(2.5)
n
X
ain yi un
n−1
X
i=1
anj xn zj −
j=1
+
n−1
X
anj yn uj
j=1
+
+
n−1
X
n−1
X
anj yn zj
j=1
n
X
n
X
i=1
i=1
n−1
X
i,j=1
i,j=1
n−1
X
n−1
X
i,j=1
aij ann yi uj xn zn −
Title Page
!
Contents
ain xi un
i=1
ain xi zn −
aij ann xi zj yn un −
n
X
ain yi zn
n−1
X
!
anj xn uj
j=1
!
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aij ann yi zj xn un
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!
aij ann xi uj yn zn
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i,j=1
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n−1
X
=
ain yi un
n−1
X
i=1
+
n−1
X
anj yn uj
j=1
+
+
anj xn zj −
j=1
n−1
X
ain xi zn −
i=1
n−1 X
n−1
X
anj yn zj
j=1
aij ann xi zj yn un −
n−1
X
ain yi zn
n−1 X
n−1
X
i=1 j=1
i=1 j=1
n−1 X
n−1
X
n−1 X
n−1
X
aij ann yi uj xn zn −
n−1 X
n−1
X
n−1
X
!
ain xi un
i=1
i=1
i=1 j=1
=
n−1
X
n−1
X
!
anj xn uj
j=1
!
aij ann yi zj xn un
On Certain Inequalities Related
to the Seitz Inequality
!
aij ann xi uj yn zn
Liang-Cheng Wang and Jia-Gui
Luo
i=1 j=1
ain anj yi un xn zj + xi zn yn uj − xi un yn zj − yi zn xn uj
i=1 j=1
+
n−1 X
n−1
X
aij ann xi zj yn un + xn zn yi uj − yi zj xn un − xi uj yn zn
i=1 j=1
=
n−1 X
n−1 X
aij ann − ain anj
× xi zj yn un + xn zn yi uj − yi zj xn un − xi uj yn zn
n−1 X
n−1 X
aij ain xi xn zj zn =
anj ann yi yn uj un ≥ 0.
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i=1 j=1
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By (2.1)-(2.5) and definition of G(n), we have
G(n) − G(n − 1) =
n−1
X
aij xi zj +
i,j=1
n
X
ain xi zn +
i=1
×
n−1
X
−
aij yi uj +
i,j=1
×
n
X
−
aij yi zj +
=
aij xi zj
+
n−1
X
n−1
X
n−1
X
+
i,j=1
n−1
X
!
anj yn uj
!
On Certain Inequalities Related
to the Seitz Inequality
anj xn uj
j=1
n
X
ain yi zn +
n−1
X
aij yi uj −
anj yn zj
n−1
X
aij xi uj
i,j=1
n−1
X
aij ann xi zj yn un −
Liang-Cheng Wang and Jia-Gui
Luo
!
j=1
!
n−1
X
aij yi zj
Contents
n−1
X
JJ
J
!
ain xi un
i=1
n−1
X
Title Page
i,j=1
aij yi zj
i,j=1
i,j=1
n−1
X
ain xi un +
ain yi un −
i=1
n−1
X
j=1
i,j=1
aij xi zj
i,j=1
ain yi un +
i=1
i,j=1
n−1
X
n
X
i=1
i,j=1
n−1
X
anj xn zj
i=1
aij xi uj +
n−1
X
!
j=1
i,j=1
n−1
X
n−1
X
II
I
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!
aij ann yi zj xn un
Close
i,j=1
aij yi uj
n−1
X
i=1
ain xi zn −
n−1
X
i,j=1
aij xi uj
n−1
X
!
ain yi zn
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i=1
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+
n−1
X
aij ann yi uj xn zn −
i,j=1
+
+
+
+
n−1
X
aij xi zj
n−1
X
anj yn uj −
n−1
X
n−1
X
aij xi uj
i,j=1
j=1
i,j=1
j=1
n−1
X
n−1
X
n−1
X
n−1
X
aij yi uj
anj xn zj −
aij yi zj
i,j=1
j=1
i,j=1
j=1
n
X
n
X
n
X
n
X
ain yi un
ajn xj zn −
ain yi zn
i=1
j=1
i=1
j=1
n−1
X
n−1
X
n−1
X
n−1
X
ani yn ui
n
X
n−1
X
j=1
anj xn zj −
j=1
ain yi un
i=1
+
aij ann xi uj yn zn
i,j=1
i=1
+
!
n−1
X
n−1
X
anj xn zj −
j=1
anj yn uj
n
X
i=1
ani yn zi
i=1
n−1
X
ain xi zn −
anj yn zj
!
anj xn uj
!
ajn xj un
!
anj xn uj
n
X
anj yn zj
Title Page
!
ain xi un
i=1
ain yi zn
n−1
X
i=1
≥ 0,
i.e., inequality (1.12) is true. This completes the proof of theorem .
j=1
On Certain Inequalities Related
to the Seitz Inequality
Liang-Cheng Wang and Jia-Gui
Luo
j=1
j=1
n
X
!
!
anj xn uj
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3.
Applications
Let E be a convex subset of an arbitrary real linear space K, and let f : E 7→
(0, +∞). f is an exponential convex function on E, if and only if
(3.1)
f tu + (1 − t)v ≤ f t (u)f 1−t (v)
for any u, v ∈ E and any t ∈ [0, 1]. f is an exponential concave function on E,
if and only if the inequality (3.1) reverses (see [4]).
For any u, v ∈ E(u 6= v) and αki , βki ∈ [0, 1], we let xki = αki u +(1− αki )v
and yki = βki u + (1 − βki )v (k = 1, 2; i = 1, 2, . . . , n; n > 2). Define a
function L by
L(n) =
n
X
aij f (x1i )f (x2j )
i,j=1
n
X
aij f (y1i )f (y2j )
−
aij f (x1i )f (y2j )
i,j=1
n
X
Contents
aij f (y1i )f (x2j ).
i,j=1
Proposition 3.1. Let f be an exponential convex (or concave) function on E
and inequality (1.8) be true. For k = 1, 2 and every pair of positive integers i
and j such that 1 ≤ i < j ≤ n, if
(3.2)
αki ≤ βki ≤ αkj
and
αkj − αki = βkj − βki ,
then we have
(3.3)
Liang-Cheng Wang and Jia-Gui
Luo
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i,j=1
n
X
On Certain Inequalities Related
to the Seitz Inequality
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L(n) ≥ L(n − 1).
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Proof. 1. Suppose f is an exponential convex function on E. For k = 1, 2
and 1 ≤ i < j ≤ n, from (3.2), we have βkj = αkj + βki − αki ≥ αkj .
Case 1. When αki < βki ≤ αkj < βkj , we take t =
βkj −βki
.
βkj −αki
(3.4)
βki −αki
,
βkj −αki
then 1 − t =
Hence, we have
tykj + (1 − t)xki = βki u + (1 − βki )v = yki .
From (3.1) and (3.4), we have
f (yki ) ≤ f t (ykj )f 1−t (xki ).
(3.5)
From (3.2), we get the other form of t and 1 − t: t =
αkj −αki
.
βkj −αki
(3.6)
On Certain Inequalities Related
to the Seitz Inequality
βkj −αkj
βkj −αki
and 1 − t =
Then we have
Title Page
(1 − t)ykj + txki = αkj u + (1 − αkj )v = xkj .
From (3.1) and (3.6), we have
(3.7)
Liang-Cheng Wang and Jia-Gui
Luo
f (xkj ) ≤ f 1−t (ykj )f t (xki ).
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From (3.5) and (3.7), we obtain
f (xki ) f (xkj )
(3.8)
f (yki ) f (ykj )
Close
≥ 0.
Case 2. When αki = βki (or αkj = βkj ), by (3.2), then we have αkj = βkj
(or αki = βki ). Hence, the equality of (3.8) holds.
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For any 1 ≤ i < j ≤ n and any 1 ≤ r < s ≤ n, by (3.8), we obtain
f (x1i ) f (x1j ) f (x2r ) f (x2s ) (3.9)
f (y1i ) f (y1j ) f (y2r ) f (y2s ) ≥ 0.
2. Let f be an exponential concave function on E. Then (3.5), (3.7) and (3.8)
reverse. Hence, (3.9) is still valid.
Replacing xi , yi , zi and ui in Theorem 1.1 with f (x1i ), f (y1i ), f (x2i ) and
f (y2i )(i = 1, 2, . . . , n), respectively, we obtain (3.3). This completes the
proof of Proposition 3.1.
Remark 3.1. In Proposition 3.1, when E is a real interval I, we only need
xki ≤ yki ≤ xkj
and
xkj − xki = ykj − yki ,
where k = 1, 2, i, j are every pair of positive integers such that 1 ≤ i < j ≤ n,
xki , xkj , yki , ykj ∈ I.
In order to verify Proposition 3.3, the following lemma is necessary.
Lemma 3.2. Let c, d : [a, b] 7→ R (b > a) be the monotonic functions, both
increasing or both decreasing. Furthermore, let q : [a, b] 7→ (0, +∞) be an
integrable function. Then
Z b
Z b
Z b
Z b
(3.10)
q(x)c(x)dx
q(x)d(x)dx ≤
q(x)dx
q(x)c(x)d(x)dx.
a
a
a
a
If one of the functions of c or d is increasing and the other decreasing, then the
inequality (3.10) reverses. (see [2, 3]).
On Certain Inequalities Related
to the Seitz Inequality
Liang-Cheng Wang and Jia-Gui
Luo
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Let p : [a, b] 7→ (0, +∞) be continuous, g : [a, b] 7→ (1, +∞) be monotonic
continuous. Define a function M by
M (n) =
n
X
(k+i)
aij h
(m+j)
(x)h
i,j
(x)
n
X
aij h(l+i) (x)h(w+j) (x)
i,j
−
n
X
aij h(k+i) (x)h(w+j) (x)
i,j
n
X
aij h(l+i) (x)h(m+j) (x),
i,j
where k, l, m, w ∈ N, i, j = 1, 2, . . . , n and
Z b
x
+
(3.11)
h : R 7→ R ,
h(x) =
p(t) g(t) dt (see [5]).
On Certain Inequalities Related
to the Seitz Inequality
Liang-Cheng Wang and Jia-Gui
Luo
a
Proposition 3.3. Let the inequalities in (1.8) hold. If k < l, m < w or k > l,
m > w, then we have
(3.12)
M (n) ≥ M (n − 1).
Proof. For (3.11), by continuity of p and g, we may change the order of derivation and integration. By direct computation, we get
Z b
(n)
(3.13)
h (x) =
p(t) (g(t))x (ln g(t))n dt.
a
For every pair of integers i, j such that 1 ≤ i < j ≤ n, when k < l, replace q,
c and d in Lemma 3.2 by p(t) (g(t))x (ln g(t))k+i , (ln g(t))j−i and (ln g(t))l−k ,
respectively. Using (3.13), we get
(3.14)
h(k+i) (x)h(l+j) (x) ≥ h(k+j) (x)h(l+i) (x).
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By (3.14), we have
(3.15)
(k+i)
h
(x) h(k+j) (x)
h(l+i) (x) h(l+j) (x)
≥ 0.
(m+r)
h
(x) h(m+s) (x)
h(w+r) (x) h(w+s) (x)
≥ 0,
Similarly we obtain
(3.16)
where r, s are pair of integer such that 1 ≤ r < s ≤ n and m < w.
Replacing xi , yi , zi and ui in Theorem 1.1 by h(k+i) (x), h(l+i) (x), h(m+i) (x)
and h(w+i) (x)(i = 1, 2, . . . , n), respectively, we obtain (3.12).
By Lemma 3.2, when k > l and m > w, both (3.15) and (3.16) reverse.
Hence, the product on the left for both (3.15) and (3.16) is still nonnegative,
hence, by Theorem 1.1, (3.12) is also satisfied.
This completes the proof of Proposition 3.3.
On Certain Inequalities Related
to the Seitz Inequality
Liang-Cheng Wang and Jia-Gui
Luo
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References
[1] G. SEITZ, Une remarque aux inégalités, Aktuarské Vědy, 6 (1936), 167–
171.
[2] D.S. MITRINOVIĆ, Analytic Inequalities, Springer-Verlag, Berlin/New
York, 1970.
[3] BAI-NI GUO AND FENG QI, Inequalities for generalized weighted mean
values of convex function, Math. Ineq. Appl., 4(2) (2001), 195–202.
[4] CHUNG-LIE WANG, Inequalities of the Rado-Popoviciu type for functions
and their applications, J. Math. Anal. Appl., 100 (1984), 436–446.
[5] FENG QI, Generalized weighted mean values with two parameters , Proc.
R. Soc. Lond. A., 454 (1998), 2723–2732.
[6] P.M. VASIĆ AND J.E. PEČARIĆ, On the Jensen inequality, Univ. Beograd.
Publ. Elektrotehn. Fak. Ser. Math. Fiz., 634–677 (1979), 50–54.
[7] L.-C. WANG, On the monotonicity of difference generated by the inequality of Chebyshev type, J. Sichuan University (Natural Science Edition), 39
(2002), 338–403.
[8] L.-C. WANG, Convex Functions and Their Inequalities, Sichuan University
Press, Chengdu, China, 2001. (In Chinese).
On Certain Inequalities Related
to the Seitz Inequality
Liang-Cheng Wang and Jia-Gui
Luo
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