ON AN INEQUALITY OF V. CSISZÁR AND T.F.
MÓRI FOR CONCAVE FUNCTIONS OF TWO
VARIABLES
BOŽIDAR IVANKOVIĆ
SAICHI IZUMINO
Faculty of Transport and Traffic Engineering
University of Zagreb, Vukelićeva 4
10000 Zagreb, Croatia
EMail: ivankovb@fpz.hr
Faculty of Education
Toyama University
Gofuku, Toyama 930-8555, JAPAN
EMail: s-izumino@h5.dion.ne.jp
JOSIP E. PEČARIĆ
MASARU TOMINAGA
Faculty of Textile Technology
University of Zagreb, Pierottijeva 6
10000 Zagreb, Croatia
EMail: pecaric@mahazu.hazu.hr
Toyama National College of Technology
13, Hongo-machi, Toyama-shi
939-8630, Japan
EMail: mtommy@toyama-nct.ac.jp
Inequality of V. Csiszár
and T.F. Móri
Božidar Ivanković, Saichi Izumino,
Josip E. Pečarić and Masaru Tominaga
vol. 8, iss. 3, art. 88, 2007
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Page 1 of 18
Received:
27 September, 2006
Accepted:
21 April, 2007
Communicated by:
I. Pinelis
2000 AMS Sub. Class.:
26D15.
Key words:
Diaz-Metcalf inequality, Hölder’s inequality, Hadamard’s inequality, Petrović’s inequality, Giaccardi’s inequality.
Abstract:
V. Csiszár and T.F. Móri gave an extension of Diaz-Metcalf’s inequality for concave
functions. In this paper, we show its restatement. As its applications we first give a
reverse inequality of Hölder’s inequality. Next we consider two variable versions of
Hadamard, Petrović and Giaccardi inequalities.
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Contents
1
Introduction
3
2
Reverse Hölder’s Inequality
5
3
Hadamard’s Inequality
9
4
Petrović’s Inequality
11
5
Giaccardi’s Inequality
14
Inequality of V. Csiszár
and T.F. Móri
Božidar Ivanković, Saichi Izumino,
Josip E. Pečarić and Masaru Tominaga
vol. 8, iss. 3, art. 88, 2007
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1.
Introduction
In this paper, let (X, Y ) be a random vector with P [(X, Y ) ∈ D] = 1 where D :=
[a, A] × [b, B] (0 ≤ a < A and 0 ≤ b < B). Let E[X] be the expectation of a
random variable X with respect to P . For a function φ : D → R, we put
∆φ = ∆φ(a, b, A, B) := φ(a, b) − φ(a, B) − φ(A, b) + φ(A, B).
In [1], V. Csiszár and T.F. Móri showed the following theorem as an extension of
Diaz-Metcalf’s inequality [2].
Inequality of V. Csiszár
and T.F. Móri
Božidar Ivanković, Saichi Izumino,
Josip E. Pečarić and Masaru Tominaga
vol. 8, iss. 3, art. 88, 2007
Theorem A. Let φ : D → R be a concave function.
We use the following notations:
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φ(A, b) − φ(a, b)
φ(a, B) − φ(a, b)
λ1 = λ4 :=
, µ1 = µ3 :=
,
A−a
B−b
φ(A, B) − φ(a, B)
φ(A, B) − φ(A, b)
λ2 = λ3 :=
, µ2 = µ4 :=
,
A−a
B−b
b
a
AB − ab
ν1 :=
φ(a, b) −
φ(a, B) −
φ(A, b),
(A − a)(B − b)
B−b
A−a
A
B
AB − ab
ν2 :=
φ(a, B) +
φ(A, b) −
φ(A, B),
A−a
B−b
(A − a)(B − b)
B
a
aB − Ab
ν3 :=
φ(a, b) −
φ(A, B) +
φ(a, B),
B−b
A−a
(A − a)(B − b)
A
b
aB − Ab
ν4 :=
φ(a, b) −
φ(A, B) −
φ(A, b).
A−a
B−b
(A − a)(B − b)
a) Suppose that ∆φ ≥ 0.
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and
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a − (i) If (B − b)E[X] + (A − a)E[Y ] ≤ AB − ab, then
λ1 E[X] + µ1 E[Y ] + ν1 ≤ E[φ(X, Y )] (≤ φ(E[X], E[Y ])) .
a − (ii) If (B − b)E[X] + (A − a)E[Y ] ≥ AB − ab, then
λ2 E[X] + µ2 E[Y ] + ν2 ≤ E[φ(X, Y )] (≤ φ(E[X], E[Y ])) .
b) Suppose that ∆φ ≤ 0
b − (iii) If (B − b)E[X] + (A − a)E[Y ] ≤ aB − Ab, then
λ3 E[X] + µ3 E[Y ] + ν3 ≤ E[φ(X, Y )] (≤ φ(E[X], E[Y ])) .
Inequality of V. Csiszár
and T.F. Móri
Božidar Ivanković, Saichi Izumino,
Josip E. Pečarić and Masaru Tominaga
vol. 8, iss. 3, art. 88, 2007
b − (iv) If (B − b)E[X] + (A − a)E[Y ] ≥ aB − Ab, then
λ4 E[X] + µ4 E[Y ] + ν4 ≤ E[φ(X, Y )] (≤ φ(E[X], E[Y ])) .
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Let us note that Theorem A can be given in the following form:
Theorem 1.1. Suppose that φ : D → R is a concave function.
a) If ∆φ ≥ 0, then
(1.1)
max{λk E[X] + µk E[Y ] + νk } ≤ E[φ(X, Y )](≤ φ(E[X], E[Y ]),
k=1,2
where λk , µk and νk (k = 1, 2) are defined in Theorem A.
b) If ∆φ ≤ 0,then
(1.2)
max{λk E[X] + µk E[Y ] + νk } ≤ E[φ(X, Y )](≤ φ(E[X], E[Y ]),
k=3,4
where λk , µk and νk (k = 3, 4) are defined in Theorem A.
Remark 1. The inequality E[φ(X, Y )] ≤ φ(E[X], E[Y ]) is Jensen’s inequality. So
the inequalities in Theorem A represent reverse inequalities of it.
In this note, we shall give some applications of these results.
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2.
Reverse Hölder’s Inequality
1
1
Let p, q > 1 be real numbers with p1 + 1q = 1. Then φ(x, y) := x p y q is a concave
function on (0, ∞) × (0, ∞). For 0 < a < A and 0 < b < B, ∆φ is represented as
follows:
1
1
1 1
1
1
1 1
1
1
1
1
∆φ = a p b q − a p B q − A p b q + A p B q = A p − a p
B q − b q (> 0).
Moreover, putting A = B = 1, and replacing X, Y , a and b by X p , Y q , αp and
β , respectively, in Theorem A, we have the following result:
q
Theorem 2.1. Let p, q > 1 be real numbers with p1 + 1q = 1. Let 0 < α ≤ X ≤ 1
and 0 < β ≤ Y ≤ 1.
(i) If (1 − β q )E[X p ] + (1 − αp )E[Y q ] ≤ 1 − αp β q , then
(2.1)
β(1 − α)
α(1 − β)
p
E[X
]
+
E[Y q ]
1 − αp
1 − βq
αβ(1 − αp−1 − β q−1 + αp−1 β q + αp β q−1 − αp β q )
+
≤ E[XY ].
(1 − αp )(1 − β q )
Inequality of V. Csiszár
and T.F. Móri
Božidar Ivanković, Saichi Izumino,
Josip E. Pečarić and Masaru Tominaga
vol. 8, iss. 3, art. 88, 2007
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(ii) If (1 − β q )E[X p ] + (1 − αp )E[Y q ] ≥ 1 − αp β q , then
(2.2)
1−β
1 − α − β + αβ q + αp β − αp β q
1−α
p
q
E[X
]+
E[Y
]−
≤ E[XY ].
1 − αp
1 − βq
(1 − αp )(1 − β q )
By Theorem 2.1 we have the following inequality related to Hölder’s inequality:
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Theorem 2.2. Let p, q > 1 be real numbers with
0 < β ≤ Y ≤ 1, then
1
1
1
1
p
1
+
1
q
= 1. If 0 < α ≤ X ≤ 1 and
1
1
p p q q (β − αβ q ) p (α − αp β) q E[X p ] p E[Y q ] q
≤ (β − αβ q )E[X p ] + (α − αp β)E[Y q ]
≤ (1 − αp β q )E[XY ].
(2.3)
Proof. We have by Young’s inequality
(β − αβ q )E[X p ] + (α − αp β)E[Y q ]
1
1
= · p(β − αβ q )E[X p ] + · q(α − αp β)E[Y q ]
p
q
1
q
1
p
Josip E. Pečarić and Masaru Tominaga
vol. 8, iss. 3, art. 88, 2007
1
1
≥ {p(β − αβ q )E[X p ]} p {q(α − αp β)E[Y q ]} q
1
p
Inequality of V. Csiszár
and T.F. Móri
Božidar Ivanković, Saichi Izumino,
1
q
1
p
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1
q
= p q (β − αβ q ) (α − αp β) E[X p ] E[Y q ] .
Hence the first inequality holds. Next, we see that
(2.4)
−γ1 :=
αβ(1 − αp−1 − β q−1 + αp−1 β q + αp β q−1 − αp β q )
≥0
(1 − αp )(1 − β q )
and
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1 − α − β + αβ q + αp β − αp β q
≥ 0.
(1 − αp )(1 − β q )
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Indeed, we have (1 − αp )(1 − β q ) > 0 and moreover by Young’s inequality
Close
γ2 :=
(2.5)
1−α
p−1
−β
q−1
+α
p−1 q
p q−1
β +α β
p q
p−1
p q
−α β
= 1 − α β − α (1 − β q ) − β q−1 (1 − αp )
1 1 p
1 1 q
p q
q
≥1−α β −
+ α (1 − β ) −
+ β (1 − αp ) = 0
p q
q p
and
1 − α − β + αβ q + αp β − αp β q
= 1 − αp β q − α(1 − β q ) − β(1 − αp )
1 1 p
1 1 q
p q
q
≥1−α β −
+ α (1 − β ) −
+ β (1 − αp ) = 0.
q p
p q
Multiplying both sides of (2.1) by γ2 and those of (2.2) by −γ1 , respectively, and
taking the sum of the two inequalities, we have
α(1−β) γ β(1−α) γ 1
q
1−αp
1 E[Y q ] ≤ (γ2 − γ1 )E[XY ].
E[X p ] + 1−β
1−β
1−α
1−β q γ2 1−αp γ2 Here we note that from (2.4) and (2.5),
β(1−α) γ q−1
1−αp
1 = β(1 − α)(1 − β)(1 − αβ ) ,
1−α
(1 − αp )(1 − β q )
1−αp γ2 α(1−β) γ p−1
1 1−β q
= α(1 − α)(1 − β)(1 − α β)
1−β
(1 − αp )(1 − β q )
1−β q γ2 Inequality of V. Csiszár
and T.F. Móri
Božidar Ivanković, Saichi Izumino,
Josip E. Pečarić and Masaru Tominaga
vol. 8, iss. 3, art. 88, 2007
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and
(1 − α)(1 − β)(1 − αp β q )
γ2 − γ1 =
.
(1 − αp )(1 − β q )
Hence we have
β(1 − α)(1 − β)(1 − αβ q−1 )
α(1 − α)(1 − β)(1 − αp−1 β)
p
E[X
]
+
E[Y q ]
(1 − αp )(1 − β q )
(1 − αp )(1 − β q )
Close
≤
(1 − α)(1 − β)(1 − αp β q )
E[XY ]
(1 − αp )(1 − β q )
and so the second inequality of (2.3) holds.
The second inequality is given in [5, p.124]. In (2.3), the first and the third terms
yield the following Gheorghiu inequality [4, p.184], [5, p.124]:
Theorem B. Let p, q > 1 be real numbers with
0 < β ≤ Y ≤ 1, then
(2.6)
p
1
p
q
1
p
+
1
q
= 1. If 0 < α ≤ X ≤ 1 and
Josip E. Pečarić and Masaru Tominaga
vol. 8, iss. 3, art. 88, 2007
1 − αp β q
1
q
E[X ] E[Y ] ≤
1
1
Inequality of V. Csiszár
and T.F. Móri
Božidar Ivanković, Saichi Izumino,
1
1
E[XY ].
p p q q (β − αβ q ) p (α − αp β) q
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We see that (2.3) is a kind of a refinement of (2.6). Theorem B gives us the next
estimation.
Corollary 2.3. Let X = {ai } and Y = {bj } be independent discrete random
variables with distributions P (X = ai ) = wi and P (Y = bj ) = zj . Suppose
0 <Pα ≤ X P
≤ 1 and 0 < P
β ≤ P
Y ≤ 1. E[X p ], E[Y q ] and E[XY ] are given
n
n
n
p
q
by i=1 wi ai , i=1 zj bj and i=1 nj=1 wi zj ai bj , respectively. Then we have inequalities
! p1
! 1q
n X
n
n
n
X
X
X
wi zj ai bj ≤
wi api
zj bqj
i=1 j=1
i=1
≤
j=1
n X
n
X
1 − αp β q
1
1
1
1
p p q q (β − αβ q ) p (α − αp β) q
i=1 j=1
wi zj ai bj .
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3.
Hadamard’s Inequality
The following well-known inequality is due to Hadamard [5, p.11]: For a concave
function f : [a, b] → R,
Z b
f (a) + f (b)
1
a+b
(3.1)
≤
.
f (t)dt ≤ f
2
b−a a
2
Moreover, the following is an extension of the weighted version of Hadamard’s inequality by Fejér ([3], [6, p.138]): Let g be a positive integrable function on [a, b]
with g(a + t) = g(b − t) for 0 ≤ t ≤ 12 (a − b). Then
Z b
Z
Z b
a+b
f (a) + f (b) b
g(t)dt ≤
f (t)g(t)dt ≤ f
g(t)dt.
(3.2)
2
2
a
a
a
Inequality of V. Csiszár
and T.F. Móri
Božidar Ivanković, Saichi Izumino,
Josip E. Pečarić and Masaru Tominaga
vol. 8, iss. 3, art. 88, 2007
Title Page
Here we give an analogous result for a function of two variables.
Contents
Theorem 3.1. Let X and Y be independent random variables such that
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a+A
b+B
and E[Y ] =
2
2
for 0 < a ≤ X ≤ A and 0 < b ≤ Y ≤ B. If φ : D → R is a concave function, then
φ(A, b) + φ(a, B) φ(a, b) + φ(A, B)
(3.4)
min
,
≤ E[φ(X, Y )]
2
2
a+A b+B
,
.
≤φ
2
2
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I
(3.3)
E[X] =
Proof. We only have to prove the case ∆φ ≥ 0. Then with same notations as in
Theorem A we have
φ(A, b) + φ(a, B)
λ1 E[X] + µ1 E[Y ] + ν1 = λ2 E[X] + µ2 E[Y ] + ν2 =
2
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by (3.3). Since ∆φ ≥ 0, it is the same as the first expression in (3.4). Similarly
calculation for ∆φ ≤ 0 proves that the desired inequality (3.4) also holds.
We can obtain the following result as an extension of Hadamard’s inequality (3.1)
from Theorem 3.1 by letting X and Y be independent, uniformly distrbuted radom
variables on the intervals [a, A] and [b, B], respectively:
Corollary 3.2. Let 0 < a < A and 0 < b < B. If φ is a concave function, then
Z AZ B
φ(A, b) + φ(a, B) φ(a, b) + φ(A, B)
1
min
,
≤
φ(t, s) dsdt
2
2
(A − a)(B − b) a b
a+A b+B
≤φ
,
.
2
2
By Theorem 3.1, we have the following analogue of (3.2) for a function of two
variables:
Corollary 3.3. Let w : D → R be a nonnegative integrable function such that
w(s, t) = u(s)v(t) where u : [a, A] → R is an integrable function with u(s) =
RA
u(a + A − s), a u(s)ds = 1 and v : [b, B] → R is an integrable function such that
RB
v(t)dt = 1, v(t) = v(b + B − t). If φ is a concave function, then
b
Z AZ B
φ(A, b) + φ(a, B) φ(a, b) + φ(A, B)
min
,
≤
w(s, t)φ(s, t) dsdt
2
2
a
b
a+A b+B
,
.
≤φ
2
2
Inequality of V. Csiszár
and T.F. Móri
Božidar Ivanković, Saichi Izumino,
Josip E. Pečarić and Masaru Tominaga
vol. 8, iss. 3, art. 88, 2007
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4.
Petrović’s Inequality
The following is called Petrović’s inequality for a concave function f : [0, c] → R:
!
n
n
X
X
f
pi xi ≤
pi f (xi ) + (1 − Pn ) f (0),
i=1
i=1
where x = (x1 ,P
. . . , xn ) and p = (p1 , . . . , pn ) are n-tuples
Pn of nonnegative real numn
bers
such
that
p
x
≥
x
for
k
=
1,
.
.
.
,
n,
k
i=1 i i
i=1 pi xi ∈ [0, c] and Pn :=
Pn
i=1 pi (see [5, p.11] and [6]).
We give an analogous result for a function of two variables.
Theorem 4.1. Let p = (p1 , . P
. . , pn ) and q = (q1 , . . . , qnP
) be n-tuples of nonnegative
n
real numbers and put Pn := i=1 pi (> 0) and Qn := nj=1 qj (> 0). Suppose that
x = (x1 , . . . , xP
n-tuples of nonnegative real numbers
n ) and y = (y1 , . . . , yn ) are P
with 0 ≤ xk ≤ ni=1 pi xi ≤ c and 0 ≤ yk ≤ nj=1 qj yj ≤ d for k = 1, 2, . . . , n. Let
φ : [0, c] × [0, d] → R be a concave function.
a) Suppose
!
!
!
n
n
n
n
X
X
X
X
pi xi , 0 + φ 0,
qj y j .
φ(0, 0) + φ
pi xi ,
qj y j ≥ φ
i=1
a − (i) If
1
Pn
+
1
Qn
j=1
≤ 1, then
!
n
X
1
(4.1)
φ
pi xi , 0
Pn
i=1
i=1
Inequality of V. Csiszár
and T.F. Móri
Božidar Ivanković, Saichi Izumino,
Josip E. Pečarić and Masaru Tominaga
vol. 8, iss. 3, art. 88, 2007
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j=1
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n
X
1
+
φ 0,
qj y j
Qn
j=1
!
1
1
+ 1−
−
Pn Qn
Close
φ(0, 0)
n
n
1 XX
≤
pi qj φ(xi , yj ).
Pn Qn i=1 j=1
a − (ii) If
1
Pn
+
1
Qn
≥ 1, then
!
n
n
X
X
1
1
+
−1 φ
pi xi ,
qj y j
Pn Qn
i=1
j=1
!
! n
n
X
X
1
1
+ 1−
φ
pi xi , 0 + 1 −
φ 0,
qj y j
Qn
P
n
i=1
j=1
≤
n
n
1 XX
pi qj φ(xi , yj ).
Pn Qn i=1 j=1
b) Suppose
Inequality of V. Csiszár
and T.F. Móri
Božidar Ivanković, Saichi Izumino,
Josip E. Pečarić and Masaru Tominaga
vol. 8, iss. 3, art. 88, 2007
Title Page
n
X
φ(0, 0) + φ
pi xi ,
n
X
i=1
!
qj y j
≤φ
j=1
n
X
!
pi xi , 0
+ φ 0,
i=1
n
X
!
qj y j
Contents
.
j=1
b − (iii) If Pn ≥ Qn , then
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1
φ
Pn
n
X
pi xi ,
i=1
n
X
!
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qj y j
j=1
+
1
1
−
Qn Pn
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φ 0,
n
X
j=1
!
qj y j
+ 1−
1
Qn
φ(0, 0)
n
n
1 XX
≤
pi qj φ(xi , yj ).
Pn Qn i=1 j=1
Close
b − (iv) If Qn ≥ Pn , then
!
n
n
X
X
1
φ
pi xi ,
qj y j
Qn
i=1
j=1
! n
X
1
1
1
−
φ
φ(0, 0)
pi xi , 0 + 1 −
−
Qn Pn
P
n
i=1
n
n
1 XX
≤
pi qj φ(xi , yj ).
Pn Qn i=1 j=1
P
P
Proof. We put a = b = 0, A = ni=1 pi xi and B = nj=1 qj yj in Theorem A. Let
X = {ai } and Y = {bj } be independent discrete random variables with distributions
P (X = xi ) = Ppni and P (Y = yj ) = Qqin , 1 ≤ i ≤ n, respectively. So we have the
desired inequalities.
Specially, if pi = qj = 1 (i, j = 1, . . . , n) in Theorem 4.1, then we have the
following:
Inequality of V. Csiszár
and T.F. Móri
Božidar Ivanković, Saichi Izumino,
Josip E. Pečarić and Masaru Tominaga
vol. 8, iss. 3, art. 88, 2007
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Corollary 4.2. Suppose that x = (x1 , . . . , P
xn ) and y = (y1 , . . . ,P
yn ) are n-tuples of
n
nonnegative real numbers for n ≥ 2 with i=1 xi ∈ [0, c] and ni=1 yi ∈ [0, d]. If
φ : [0, c] × [0, d] → R is a concave function, then
!
!
n
n
n
n
X
X
1 XX
(4.2) φ
xi , 0 + φ 0,
yj + (n − 2) φ(0, 0) ≤
φ(xi , yj ).
n
i=1
j=1
i=1 j=1
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5.
Giaccardi’s Inequality
In 1955, Giaccardi (cf. [5, p.11]) proved the following inequality for a convex function f : [a, A] → R,
!
n
n
X
X
pi f (xi ) ≤ C · f
pi xi + D · (Pn − 1) · f (x0 ),
i=1
where
i=1
Pn
pi (xi − x0 )
C = Pi=1
n
i=1 pi xi − x0
Pn
pi xi
and D = Pn
i=1 pi xi − x0
P
for a nonnegative n-tuple p = (p1 , . . . , pn ) with Pn := ni=1 pi and a real (n + 1)tuple x = (x0 , x1 , . . . , xn ) such that for k = 0, 1, . . . , n
!
n
X
pi xi − x0 ≥ 0,
a ≤ xi ≤ A, (xk − x0 )
a<
n
X
pi xi < A
i=1
and
i=1
i=1
n
X
pi xi 6= x0 .
Inequality of V. Csiszár
and T.F. Móri
Božidar Ivanković, Saichi Izumino,
Josip E. Pečarić and Masaru Tominaga
vol. 8, iss. 3, art. 88, 2007
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In this section, we discuss a generalization of Giaccardi’s inequality to a function of two variables under similar conditions. Let x = (x0 , x1 , . . . , xn ) and y =
(y0 , y1 , . . . , yn ) be nonnegative (n + 1)-tuples, and p = (p1 , p2 , . . . , pn ) and q =
(q1 , q2 , . . . , qn ) be nonnegative n-tuples with
(5.1)
x0 ≤ xk ≤
n
X
i=1
pi xi
and y0 ≤ yk ≤
n
X
j=1
qj y j
for k = 1, . . . , n.
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We use the following notations:
Pn :=
n
X
pi (≥ 0),
Qn :=
i=1
qj (≥ 0),
j=1
Pn
j=1 qj yj − Qn y0
K(Y ) := Pn
,
j=1 qj yj − y0
P
(Qn − 1) nj=1 qj yj
L(Y ) := Pn
,
j=1 qj yj − y0
Pn
i=1 pi xi − Pn x0
K(X) := P
,
n
i=1 pi xi − x0
P
(Pn − 1) ni=1 pi xi
,
L(X) := Pn
i=1 pi xi − x0
(
M (X, Y ) :=
n
X
(Pn Qn − Pn − Qn )
n
X
pi xi
n
X
i=1
n
X
+ Qn y0
Inequality of V. Csiszár
and T.F. Móri
Božidar Ivanković, Saichi Izumino,
Josip E. Pečarić and Masaru Tominaga
vol. 8, iss. 3, art. 88, 2007
qj y j
Title Page
j=1
p i x i + Pn x 0
i=1
n
X
Contents
)
qj yj − Pn Qn x0 y0
j=1
× P
( ni=1 pi xi − x0 )
1
P
n
j=1 qj yj
− y0
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and
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N (X, Y )
(Pn − Qn )
:=
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n
P
i=1
pi xi
n
P
qj yj − (Pn − 1)Qn
j=1
n
P
i=1
n
P
i=1
pi xi − x0
n
P
j=1
n
P
pi xi y0 + Pn (Qn − 1)x0
qj y j
j=1
!
.
qj y j − y 0
Then we have the following theorem:
P
P
Theorem 5.1. Let φ : [x0 , ni=1 pi xi ] × [y0 , nj=1 qj yj ] → R be a concave function.
a) If
!
!
!
n
n
n
n
X
X
X
X
φ(x0 , y0 ) + φ
pi xi ,
qj y j ≥ φ x 0 ,
qj y j + φ
p i x i , y0 ,
i=1
j=1
j=1
Inequality of V. Csiszár
and T.F. Móri
Božidar Ivanković, Saichi Izumino,
i=1
then
Josip E. Pečarić and Masaru Tominaga
(
max Qn K(X)φ
n
X
!
p i x i , y0
+ Pn K(Y )φ x0 ,
i=1
Pn L(Y )φ
vol. 8, iss. 3, art. 88, 2007
!
n
X
qj y j
+ M (X, Y )φ(x0 , y0 ) ,
j=1
n
X
!
p i x i , y0
+ Qn L(X)φ x0 ,
n
X
i=1
Title Page
!
qj y j
Contents
j=1
−M (X, Y )φ
n
X
i=1
pi xi ,
n
X
!)
≤
qj y j
j=1
n X
n
X
pi qj φ(xi , yj ).
i=1 j=1
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Page 16 of 18
b) If
φ(x0 , y0 ) + φ
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n
X
i=1
pi xi ,
n
X
j=1
!
qj y j
≤ φ x0 ,
n
X
j=1
!
qj y j
+φ
n
X
i=1
!
p i x i , y0
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then
(
max Qn K(X)φ
n
X
pi xi ,
i=1
n
X
!
qj y j
j=1
+ Pn L(Y )φ (x0 , y0 ) + N (X, Y )φ x0 ,
n
X
!
qj y j
,
Inequality of V. Csiszár
and T.F. Móri
Božidar Ivanković, Saichi Izumino,
j=1
Pn K(Y )φ
n
X
i=1
pi xi ,
n
X
!
qj y j
+ Qn L(X)φ (x0 , y0 )
j=1
−N (X, Y )φ
n
X
i=1
!)
p i x i , y0
≤
n X
n
X
pi qj φ(xi , yj ).
Josip E. Pečarić and Masaru Tominaga
vol. 8, iss. 3, art. 88, 2007
Title Page
i=1 j=1
Contents
Proof.PLet X and Y be as they were
Pn in the proof of Theorem 4.1, and put a = x0 ,
n
A = i=1 pi xi , b = y0 and B = j=1 qj yj , and use Theorem A. Then we have the
desired inequalities of this theorem.
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References
[1] V. CSISZÁR AND T.F. MÓRI, The convexity method of proving moment-type
inequalities, Statist. Probab. Lett., 66 (2004), 303–313.
[2] J.B. DIAZ AND F.T. METCALF, Stronger forms of a class of inequalities of
G. Pólya-G. Szegö, and L. V. Kantorovich, Bull. Amer. Math. Soc., 69 (1963),
415–418.
[3] L. FEJÉR, Über die Fourierreihen, II., Math. Naturwiss, Ant. Ungar. Acad. Wiss,
24 (1906), 369–390 (in Hungarian).
[4] S. IZUMINO AND M. TOMINAGA, Estimations in Hölder’s type inequalities,
Math. Inequal. Appl., 4 (2001), 163–187.
[5] D. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Classical and New Inequalities in Analysis, Kluwer. Acad. Pub., Boston, London 1993.
[6] J.E. PEČARIĆ, F. PROSCHAN AND Y.L. TONG, Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering,
Georgia Institute of Technology, 1992.
Inequality of V. Csiszár
and T.F. Móri
Božidar Ivanković, Saichi Izumino,
Josip E. Pečarić and Masaru Tominaga
vol. 8, iss. 3, art. 88, 2007
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Contents
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