Volume 8 (2007), Issue 3, Article 88, 10 pp.
ON AN INEQUALITY OF V. CSISZÁR AND T.F. MÓRI FOR CONCAVE
FUNCTIONS OF TWO VARIABLES
BOŽIDAR IVANKOVIĆ, SAICHI IZUMINO, JOSIP E. PEČARIĆ, AND MASARU TOMINAGA
FACULTY OF T RANSPORT AND T RAFFIC E NGINEERING
U NIVERSITY OF Z AGREB , V UKELI ĆEVA 4
10000 Z AGREB , C ROATIA
ivankovb@fpz.hr
FACULTY OF E DUCATION
T OYAMA U NIVERSITY
G OFUKU , T OYAMA 930-8555, JAPAN
s-izumino@h5.dion.ne.jp
FACULTY OF T EXTILE T ECHNOLOGY
U NIVERSITY OF Z AGREB , P IEROTTIJEVA 6
10000 Z AGREB , C ROATIA
pecaric@mahazu.hazu.hr
T OYAMA NATIONAL C OLLEGE OF T ECHNOLOGY
13, H ONGO - MACHI , T OYAMA - SHI
939-8630, JAPAN
mtommy@toyama-nct.ac.jp
Received 27 September, 2006; accepted 21 April, 2007
Communicated by I. Pinelis
A BSTRACT. 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.
Key words and phrases: Diaz-Metcalf inequality, Hölder’s inequality, Hadamard’s inequality, Petrović’s inequality, Giaccardi’s inequality.
2000 Mathematics Subject Classification. 26D15.
1. I NTRODUCTION
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).
127-07
2
B. I VANKOVI Ć , S. I ZUMINO , J.E. P E ČARI Ć , AND M. T OMINAGA
In [1], V. Csiszár and T.F. Móri showed the following theorem as an extension of DiazMetcalf’s inequality [2].
Theorem A. Let φ : D → R be a concave function.
We use the following notations:
φ(A, b) − φ(a, b)
φ(a, B) − φ(a, b)
, 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
AB − ab
b
a
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)
λ1 = λ4 :=
and
a) Suppose that ∆φ ≥ 0.
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 ])) .
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 ])) .
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 1.2. 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.
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 88, 10 pp.
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O N AN INEQUALITY OF V. C SISZÁR AND T.F. M ÓRI FOR CONCAVE FUNCTIONS
3
2. R EVERSE H ÖLDER ’ S I NEQUALITY
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
p
q
p
q
p
q
p
q
p
p
q
q
∆φ = a b − a B − A b + A B = A − a
B −b
(> 0).
Moreover, putting A = B = 1, and replacing X, Y , a and b by X p , Y q , αp and β q , respectively, in Theorem A, we have the following result:
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 )
(ii) If (1 − β q )E[X p ] + (1 − αp )E[Y q ] ≥ 1 − αp β q , then
(2.2)
1−α
1−β
1 − α − β + αβ q + αp β − αp β q
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:
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
(2.3)
≤ (β − αβ q )E[X p ] + (α − αp β)E[Y q ]
≤ (1 − αp β q )E[XY ].
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
1
≥ {p(β − αβ q )E[X p ]} p {q(α − αp β)E[Y q ]} q
1
1
1
1
1
1
= p p q q (β − αβ q ) p (α − αp β) q E[X p ] p E[Y q ] q .
Hence the first inequality holds. Next, we see that
(2.4)
αβ(1 − αp−1 − β q−1 + αp−1 β q + αp β q−1 − αp β q )
−γ1 :=
≥0
(1 − αp )(1 − β q )
and
(2.5)
γ2 :=
1 − α − β + αβ q + αp β − αp β q
≥ 0.
(1 − αp )(1 − β q )
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 88, 10 pp.
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4
B. I VANKOVI Ć , S. I ZUMINO , J.E. P E ČARI Ć , AND M. T OMINAGA
Indeed, we have (1 − αp )(1 − β q ) > 0 and moreover by Young’s inequality
1 − αp−1 − β q−1 + αp−1 β q + αp β q−1 − αp β q
= 1 − αp β q − αp−1 (1 − β q ) − β q−1 (1 − αp )
1 1 q
1 1 p
q
p 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
β(1−α) γ α(1−β) γ 1 q
1−αp
1 E[X p ] + 1−β
E[Y q ] ≤ (γ2 − γ1 )E[XY ].
1−β
1−α
1−αp γ2 1−β q 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 and
γ2 − γ1 =
(1 − α)(1 − β)(1 − αp β q )
.
(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 )
(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
Y ≤ 1, then
(2.6)
1
+
1
q
= 1. If 0 < α ≤ X ≤ 1 and 0 < β ≤
1 − αp β q
1
E[X p ] p E[Y q ] q ≤
1
p
1
p
1
q
1
1
E[XY ].
p q (β − αβ q ) p (α − αp β) q
We see that (2.3) is a kind of a refinement of (2.6). Theorem B gives us the next estimation.
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 88, 10 pp.
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O N AN INEQUALITY OF V. C SISZÁR AND T.F. M ÓRI FOR CONCAVE FUNCTIONS
5
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
α ≤ X ≤ 1
Pn 0 <
p Pn
q
p
q
and
0
<
β
≤
Y
≤
1.
E[X
],
E[Y
]
and
E[XY
]
are
given
by
w
a
,
i=1 i i
i=1 zj bj and
Pn Pn
i=1
j=1 wi zj ai bj , respectively. Then we have inequalities
n X
n
X
wi zj ai bj ≤
n
X
i=1 j=1
! p1
wi api
i=1
≤
n
X
! 1q
zj bqj
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
wi zj ai bj .
i=1 j=1
3. H ADAMARD ’ S I NEQUALITY
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 ≤ 21 (a − b). Then
Z b
Z
Z b
f (a) + f (b) b
a+b
(3.2)
g(t)dt ≤
f (t)g(t)dt ≤ f
g(t)dt.
2
2
a
a
a
Here we give an analogous result for a function of two variables.
Theorem 3.1. Let X and Y be independent random variables such that
(3.3)
E[X] =
a+A
2
and
E[Y ] =
b+B
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
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
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:
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 88, 10 pp.
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6
B. I VANKOVI Ć , S. I ZUMINO , J.E. P E ČARI Ć , AND M. T OMINAGA
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) =
RA
u(s)v(t) where u : [a, A] → R is an integrable function with u(s) = u(a + A − s), a u(s)ds =
RB
1 and v : [b, B] → R is an integrable function such that b v(t)dt = 1, v(t) = v(b + B − t). If
φ is a concave function, then
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
4. P ETROVI Ć ’ S I NEQUALITY
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
. , pn ) are n-tuples of nonnegative
Pxn = (x1 , . . . , xn ) and p = (p1 , . .P
Pn real numbers such
n
that i=1 pi xi ≥ xk for k = 1, . . . , n, i=1 pi xi ∈ [0, c] and 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 pP
= (p1 , . . . , pn ) and q = (qP
1 , . . . , qn ) be n-tuples of nonnegative real numbers and put Pn := ni=1 pi (> 0) and Qn := nj=1 qj (> 0). Suppose that x = (x1 , . . . , xn )
P
and y = (y1 , . . P
. , yn ) are n-tuples of nonnegative real numbers 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
φ(0, 0) + φ
pi xi ,
qj y j ≥ φ
pi xi , 0 + φ 0,
qj y j .
i=1
a − (i) If
1
Pn
+
1
Qn
j=1
i=1
j=1
≤ 1, then
!
! n
n
X
X
1
1
1
1
(4.1)
φ
pi xi , 0 +
φ 0,
qj y j + 1 −
−
φ(0, 0)
Pn
Q
P
Q
n
n
n
i=1
j=1
≤
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 88, 10 pp.
n
n
1 XX
pi qj φ(xi , yj ).
Pn Qn i=1 j=1
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O N AN INEQUALITY OF V. C SISZÁR AND T.F. M ÓRI FOR CONCAVE FUNCTIONS
a − (ii) If
1
Pn
+
1
Qn
7
≥ 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
Pn
i=1
j=1
n
n
1 XX
≤
pi qj φ(xi , yj ).
Pn Qn i=1 j=1
b) Suppose
φ(0, 0) + φ
n
X
n
X
pi xi ,
i=1
!
≤φ
qj y j
n
X
j=1
!
pi xi , 0
+ φ 0,
i=1
n
X
!
qj y j
.
j=1
b − (iii) If Pn ≥ Qn , then
n
n
X
X
1
φ
pi xi ,
qj y j
Pn
i=1
j=1
!
+
1
1
−
Qn Pn
φ 0,
n
X
!
1
+ 1−
Qn
qj y j
j=1
φ(0, 0)
n
n
1 XX
pi qj φ(xi , yj ).
≤
Pn Qn i=1 j=1
b − (iv) If Qn ≥ Pn , then
n
n
X
X
1
φ
pi xi ,
qj y j
Qn
i=1
j=1
!
−
1
1
−
Qn Pn
φ
n
X
i=1
!
pi xi , 0
1
+ 1−
Pn
φ(0, 0)
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:
Corollary 4.2. Suppose that xP= (x1 , . . . , xn ) and P
y = (y1 , . . . , yn ) are n-tuples of nonnegative
n
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
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 88, 10 pp.
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8
B. I VANKOVI Ć , S. I ZUMINO , J.E. P E ČARI Ć , AND M. T OMINAGA
5. G IACCARDI ’ S I NEQUALITY
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
i=1
where
Pn
pi (xi − x0 )
C = Pi=1
n
i=1 pi xi − x0
Pn
pi xi
i=1 pi xi − x0
for a nonnegative n-tuple p = (p1 , . . . , pn ) with Pn :=
(x0 , x1 , . . . , xn ) such that for k = 0, 1, . . . , n
a ≤ xi ≤ A,
a<
n
X
(xk − x0 )
pi xi < A
i=1
and D = Pn
n
X
Pn
i=1
pi and a real (n + 1)-tuple x =
!
pi xi − x0
≥ 0,
i=1
n
X
pi xi 6= x0 .
and
i=1
i=1
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
pi xi
and y0 ≤ yk ≤
i=1
n
X
qj y j
for k = 1, . . . , n.
j=1
We use the following notations:
Pn :=
n
X
pi (≥ 0),
Qn :=
i=1
(
(Pn Qn − Pn − Qn )
n
X
i=1
+ Qn y0
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
n
X
pi xi
n
X
p i x i + Pn x 0
i=1
qj y j
j=1
n
X
)
qj yj − Pn Qn x0 y0
j=1
1
P
× P
n
n
( i=1 pi xi − x0 )
j=1 qj yj − y0
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 88, 10 pp.
http://jipam.vu.edu.au/
O N AN INEQUALITY OF V. C SISZÁR AND T.F. M ÓRI FOR CONCAVE FUNCTIONS
9
and
n
P
(Pn − Qn )
n
P
pi xi
i=1
N (X, Y ) :=
n
P
pi xi y0 + Pn (Qn − 1)x0
qj y j
i=1
j=1
!
.
n
P
qj y j − y 0
qj yj − (Pn − 1)Qn
j=1
n
P
pi xi − x0
i=1
n
P
j=1
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
i=1
!
n
X
then
(
n
X
max Qn K(X)φ
p i x i , y0
+ Pn K(Y )φ x0 ,
i=1
Pn L(Y )φ
n
X
qj y j
+ M (X, Y )φ(x0 , y0 ) ,
j=1
!
p i x i , y0
!
+ Qn L(X)φ x0 ,
i=1
n
X
!
n
X
− M (X, Y )φ
qj y j
j=1
≤
pi xi ,
i=1
n
n
XX
n
X
!)
qj y j
j=1
pi qj φ(xi , yj ).
i=1 j=1
b) If
φ(x0 , y0 ) + φ
n
X
pi xi ,
i=1
n
X
!
qj y j
≤ φ x0 ,
n
X
j=1
!
qj y j
+φ
j=1
n
X
!
p i x i , y0
,
i=1
then
(
max Qn K(X)φ
n
X
pi xi ,
i=1
Pn K(Y )φ
n
X
i=1
pi xi ,
n
X
!
qj y j
+ Pn L(Y )φ (x0 , y0 ) + N (X, Y )φ x0 ,
j=1
n
X
!
qj y j
,
j=1
!
qj y j
n
X
+ Qn L(X)φ (x0 , y0 ) − N (X, Y )φ
j=1
n
X
!)
p i x i , y0
i=1
≤
n X
n
X
pi qj φ(xi , yj ).
i=1 j=1
Proof.
they were in the proof of Theorem 4.1, and put a = x0 , A =
Pn Let X and Y be as P
n
p
x
,
b
=
y
and
B
=
0
i=1 i i
j=1 qj yj , and use Theorem A. Then we have the desired inequalities of this theorem.
R EFERENCES
[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.
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 88, 10 pp.
http://jipam.vu.edu.au/
10
B. I VANKOVI Ć , S. I ZUMINO , J.E. P E ČARI Ć ,
AND
M. T OMINAGA
[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.
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 88, 10 pp.
http://jipam.vu.edu.au/