ON AN INEQUALITY OF V. CSISZÁR AND T.F. MÓRI FOR CONCAVE FUNCTIONS OF TWO VARIABLES

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Inequality of V. Csiszár and T.F. Móri Božidar Ivankovi´c, Saichi Izumino, Josip E. Peˇcari´c and Masaru Tominaga

vol. 8, iss. 3, art. 88, 2007

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ON AN INEQUALITY OF V. CSISZÁR AND T.F.

MÓRI FOR CONCAVE FUNCTIONS OF TWO VARIABLES

BOŽIDAR IVANKOVI ´C SAICHI IZUMINO

Faculty of Transport and Traffic Engineering Faculty of Education University of Zagreb, Vukeli´ceva 4 Toyama University

10000 Zagreb, Croatia Gofuku, Toyama 930-8555, JAPAN

EMail:ivankovb@fpz.hr EMail:s-izumino@h5.dion.ne.jp

JOSIP E. PE ˇCARI ´C MASARU TOMINAGA

Faculty of Textile Technology Toyama National College of Technology University of Zagreb, Pierottijeva 6 13, Hongo-machi, Toyama-shi

10000 Zagreb, Croatia 939-8630, Japan

EMail:pecaric@mahazu.hazu.hr EMail:mtommy@toyama-nct.ac.jp

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, Petrovi´c’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, Petrovi´c and Giaccardi inequalities.

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1. Introduction

In this paper, let(X, Y)be a random vector withP[(X, Y) ∈ D] = 1whereD :=

[a, A]×[b, B] (0 ≤ a < A and 0 ≤ b < B). Let E[X] be the expectation of a random variableXwith respect toP. For a functionφ: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].

Theorem A. Letφ:D→Rbe a concave function.

We use the following notations:

λ₁ =λ₄ := φ(A, b)−φ(a, b)

A−a , µ₁ =µ₃ := φ(a, B)−φ(a, b) B−b , λ₂ =λ₃ := φ(A, B)−φ(a, B)

A−a , µ₂ =µ₄ := φ(A, B)−φ(A, b)

B−b , and ν1 := AB−ab

(A−a)(B−b)φ(a, b)− b

B −bφ(a, B)− a

A−aφ(A, b), ν₂ := A

A−aφ(a, B) + B

B−bφ(A, b)− AB−ab

(A−a)(B −b)φ(A, B), ν₃ := B

B−bφ(a, b)− a

A−aφ(A, B) + aB−Ab

(A−a)(B −b)φ(a, B), ν4 := A

A−aφ(a, b)− b

B−bφ(A, B)− aB−Ab

(A−a)(B−b)φ(A, b).

a) Suppose that∆φ ≥0.

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a−(i) If(B−b)E[X] + (A−a)E[Y]≤AB−ab, then

λ₁E[X] +µ₁E[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

λ₂E[X] +µ₂E[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

λ₃E[X] +µ₃E[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

λ4E[X] +µ4E[Y] +ν4 ≤E[φ(X, Y)] (≤φ(E[X], E[Y])). Let us note that TheoremAcan be given in the following form:

Theorem 1.1. Suppose thatφ:D→Ris a concave function.

a)If∆φ ≥0, then

(1.1) max

k=1,2{λ_kE[X] +µ_kE[Y] +ν_k} ≤E[φ(X, Y)](≤φ(E[X], E[Y]), whereλ_k, µ_kandν_k(k = 1,2) are defined in TheoremA.

b)If∆φ ≤0,then

(1.2) max

k=3,4{λ_kE[X] +µ_kE[Y] +ν_k} ≤E[φ(X, Y)](≤φ(E[X], E[Y]), whereλ_k, µ_kandν_k(k = 3,4) are defined in TheoremA.

Remark 1. The inequalityE[φ(X, Y)] ≤ φ(E[X], E[Y])is Jensen’s inequality. So the inequalities in TheoremArepresent reverse inequalities of it.

In this note, we shall give some applications of these results.

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2. Reverse Hölder’s Inequality

Letp, q > 1be real numbers with ¹_p + ¹_q = 1. Thenφ(x, y) := x¹^py¹^q is a concave function on(0,∞)×(0,∞). For0 < a < Aand0 < b < B, ∆φ is represented as follows:

∆φ=a¹^pb¹^q −a¹^pB¹^q −A^p¹b¹^q +A¹^pB¹^q =

A¹^p −a^p¹ B¹^q −b¹^q

(>0).

Moreover, puttingA =B = 1, and replacing X,Y, aandb byX^p, Y^q,α^p and β^q, respectively, in TheoremA, we have the following result:

Theorem 2.1. Letp, q > 1be real numbers with ¹_p +¹_q = 1. Let0 < α ≤ X ≤ 1 and0< β≤Y ≤1.

(i) If(1−β^q)E[X^p] + (1−α^p)E[Y^q]≤1−α^pβ^q, then (2.1) β(1−α)

1−α^p E[X^p] + α(1−β) 1−β^q E[Y^q]

+αβ(1−α^p−1−β^q−1+α^p−1β^q+α^pβ^q−1−α^pβ^q)

(1−α^p)(1−β^q) ≤E[XY].

(ii) If(1−β^q)E[X^p] + (1−α^p)E[Y^q]≥1−α^pβ^q, then (2.2) 1−α

1−α^pE[X^p]+ 1−β

1−β^qE[Y^q]−1−α−β+αβ^q+α^pβ−α^pβ^q

(1−α^p)(1−β^q) ≤E[XY].

By Theorem2.1we have the following inequality related to Hölder’s inequality:

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Theorem 2.2. Letp, q >1be real numbers with ¹_p + ¹_q = 1. If0< α≤X ≤1and 0< β≤Y ≤1, then

p¹^pq¹^q(β−αβ^q)¹^p(α−α^pβ)¹^qE[X^p]¹^pE[Y^q]¹^q (2.3)

≤(β−αβ^q)E[X^p] + (α−α^pβ)E[Y^q]

≤(1−α^pβ^q)E[XY].

Proof. We have by Young’s inequality

(β−αβ^q)E[X^p] + (α−α^pβ)E[Y^q]

= 1

p ·p(β−αβ^q)E[X^p] +1

q ·q(α−α^pβ)E[Y^q]

≥ {p(β−αβ^q)E[X^p]}¹^p{q(α−α^pβ)E[Y^q]}¹^q

=p¹^pq¹^q(β−αβ^q)¹^p(α−α^pβ)¹^qE[X^p]¹^pE[Y^q]¹^q. Hence the first inequality holds. Next, we see that

−γ₁ := αβ(1−α^p−1−β^q−1+α^p−1β^q+α^pβ^q−1−α^pβ^q) (1−α^p)(1−β^q) ≥0 (2.4)

and

γ2 := 1−α−β+αβ^q+α^pβ−α^pβ^q (1−α^p)(1−β^q) ≥0.

(2.5)

Indeed, we have(1−α^p)(1−β^q)>0and moreover by Young’s inequality 1−α^p−1−β^q−1+α^p−1β^q+α^pβ^q−1−α^pβ^q

= 1−α^pβ^q−α^p−1(1−β^q)−β^q−1(1−α^p)

≥1−α^pβ^q− 1

p+ 1 qα^p

(1−β^q)− 1

q +1 pβ^q

(1−α^p) = 0

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and

1−α−β+αβ^q+α^pβ−α^pβ^q

= 1−α^pβ^q−α(1−β^q)−β(1−α^p)

≥1−α^pβ^q− 1

q +1 pα^p

(1−β^q)− 1

p +1 qβ^q

(1−α^p) = 0.

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) 1−α^p γ₁

1−α 1−α^p γ₂

E[X^p] +

α(1−β) 1−β^q γ₁

1−β 1−β^q γ₂

E[Y^q]≤(γ₂−γ₁)E[XY].

Here we note that from (2.4) and (2.5),

β(1−α) 1−α^p γ₁

1−α 1−α^p γ₂

= β(1−α)(1−β)(1−αβ^q−1) (1−α^p)(1−β^q) ,

α(1−β) 1−β^q γ₁

1−β 1−β^q γ2

= α(1−α)(1−β)(1−α^p−1β) (1−α^p)(1−β^q) and

γ₂−γ₁ = (1−α)(1−β)(1−α^pβ^q) (1−α^p)(1−β^q) . Hence we have

β(1−α)(1−β)(1−αβ^q−1)

(1−α^p)(1−β^q) E[X^p] +α(1−α)(1−β)(1−α^p−1β) (1−α^p)(1−β^q) E[Y^q]

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≤ (1−α)(1−β)(1−α^pβ^q)

(1−α^p)(1−β^q) E[XY] 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. Letp, q > 1be real numbers with ¹_p +¹_q = 1. If0< α ≤ X ≤1and 0< β≤Y ≤1, then

(2.6) E[X^p]¹^pE[Y^q]¹^q ≤ 1−α^pβ^q

p¹^pq¹^q(β−αβ^q)¹^p(α−α^pβ)¹^q

E[XY].

We see that (2.3) is a kind of a refinement of (2.6). TheoremBgives us the next estimation.

Corollary 2.3. Let X = {a_i} and Y = {b_j} be independent discrete random variables with distributions P(X = a_i) = w_i and P(Y = b_j) = z_j. Suppose 0 < α ≤ X ≤ 1 and 0 < β ≤ Y ≤ 1. E[X^p], E[Y^q] and E[XY] are given byPn

i=1w_ia^p_i,Pn

i=1z_jb^q_j and Pn i=1

Pn

j=1w_iz_ja_ib_j, respectively. Then we have in- equalities

n

X

i=1 n

X

j=1

w_iz_ja_ib_j ≤

n

X

i=1

w_ia^p_i

!¹_p _n X

j=1

z_jb^q_j

!¹_q

≤ 1−α^pβ^q

p^p¹q¹^q(β−αβ^q)¹^p(α−α^pβ)¹^q

n

X

i=1 n

X

j=1

w_iz_ja_ib_j.

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3. Hadamard’s Inequality

The following well-known inequality is due to Hadamard [5, p.11]: For a concave functionf : [a, b]→R,

(3.1) f(a) +f(b)

2 ≤ 1

b−a Z b

a

f(t)dt ≤f

a+b 2

.

Moreover, the following is an extension of the weighted version of Hadamard’s inequality by Fejér ([3], [6, p.138]): Letg be a positive integrable function on[a, b]

withg(a+t) = g(b−t)for0≤t≤ ¹₂(a−b). Then (3.2) f(a) +f(b)

2

Z b

a

g(t)dt≤ Z b

a

f(t)g(t)dt≤f

a+b 2

Z b

a

g(t)dt.

Here we give an analogous result for a function of two variables.

Theorem 3.1. LetX andY be independent random variables such that

(3.3) E[X] = a+A

2 and E[Y] = b+B 2

for0< a≤X ≤Aand0< b ≤Y ≤B. Ifφ :D→Ris a concave function, then min

φ(A, b) +φ(a, B)

2 ,φ(a, b) +φ(A, B) 2

≤E[φ(X, Y)]

(3.4)

≤φ

a+A

2 ,b+B 2

. Proof. We only have to prove the case ∆φ ≥ 0. Then with same notations as in TheoremAwe have

λ₁E[X] +µ₁E[Y] +ν₁ =λ₂E[X] +µ₂E[Y] +ν₂ = φ(A, b) +φ(a, B) 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≤0proves that the desired inequality (3.4) also holds.

We can obtain the following result as an extension of Hadamard’s inequality (3.1) from Theorem3.1by lettingX andY be independent, uniformly distrbuted radom variables on the intervals[a, A]and[b, B], respectively:

Corollary 3.2. Let0< a < Aand0< b < B. Ifφis a concave function, then min

φ(A, b) +φ(a, B)

2 ,φ(a, b) +φ(A, B) 2

≤ 1

(A−a)(B−b) Z A

a

Z B

b

φ(t, s)dsdt

≤φ

a+A

2 ,b+B 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) = u(a+A−s),RA

a u(s)ds= 1andv : [b, B]→Ris an integrable function such that RB

b v(t)dt= 1,v(t) =v(b+B −t). Ifφis a concave function, then min

φ(A, b) +φ(a, B)

2 ,φ(a, b) +φ(A, B) 2

≤ Z A

a

Z B

b

w(s, t)φ(s, t)dsdt

≤φ

a+A

2 ,b+B 2

.

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4. Petrovi´c’s Inequality

The following is called Petrovi´c’s inequality for a concave functionf : [0, c]→R: f

n

X

i=1

p_ix_i

!

≤

n

X

i=1

p_if(x_i) + (1−P_n)f(0),

wherex= (x₁, . . . , x_n)andp= (p₁, . . . , p_n)aren-tuples of nonnegative real numbers such that Pn

i=1p_ix_i ≥ x_k for k = 1, . . . , n, Pn

i=1p_ix_i ∈ [0, c] and P_n :=

Pn

i=1p_i(see [5, p.11] and [6]).

We give an analogous result for a function of two variables.

Theorem 4.1. Letp= (p₁, . . . , p_n)andq= (q₁, . . . , q_n)ben-tuples of nonnegative real numbers and putP_n :=Pn

i=1p_i (>0)andQ_n :=Pn

j=1q_j (>0). Suppose that x = (x1, . . . , xn) and y = (y1, . . . , yn) are n-tuples of nonnegative real numbers with0≤xk ≤Pn

i=1pixi ≤cand0≤yk ≤Pn

j=1qjyj ≤dfork = 1,2, . . . , n. Let φ: [0, c]×[0, d]→Rbe a concave function.

a) Suppose φ(0,0) +φ

n

X

i=1

p_ix_i,

n

X

j=1

q_jy_j

!

≥φ

n

X

i=1

p_ix_i,0

!

+φ 0,

n

X

j=1

q_jy_j

! . a−(i) If _P¹

n + _Q¹

n ≤1, then (4.1) 1

P_nφ

n

X

i=1

pixi,0

! + 1

Q_nφ 0,

n

X

j=1

qjyj

! +

1− 1

P_n − 1 Q_n

φ(0,0)

≤ 1 P_nQ_n

n

X

i=1 n

X

j=1

p_iq_jφ(x_i, y_j).

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a−(ii) If _P¹

n +_Q¹

n ≥1, then 1

P_n + 1 Q_n −1

φ

n

X

i=1

p_ix_i,

n

X

j=1

q_jy_j

!

+

1− 1 Q_n

φ

n

X

i=1

p_ix_i,0

! +

1− 1

P_n

φ 0,

n

X

j=1

q_jy_j

!

≤ 1 P_nQ_n

n

X

i=1 n

X

j=1

p_iq_jφ(x_i, y_j).

b) Suppose φ(0,0) +φ

n

X

i=1

p_ix_i,

n

X

j=1

q_jy_j

!

≤φ

n

X

i=1

p_ix_i,0

!

+φ 0,

n

X

j=1

q_jy_j

! .

b−(iii) IfPn ≥Qn, then 1

P_nφ

n

X

i=1

p_ix_i,

n

X

j=1

q_jy_j

!

+ 1

Q_n − 1 P_n

φ 0,

n

X

j=1

q_jy_j

! +

1− 1

Q_n

φ(0,0)

≤ 1 P_nQ_n

n

X

i=1 n

X

j=1

p_iq_jφ(x_i, y_j).

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b−(iv) IfQ_n ≥P_n, then 1

Q_nφ

n

X

i=1

p_ix_i,

n

X

j=1

q_jy_j

!

− 1

Q_n − 1 P_n

φ

n

X

i=1

p_ix_i,0

! +

1− 1

P_n

φ(0,0)

≤ 1 PnQn

n

X

i=1 n

X

j=1

p_iq_jφ(x_i, y_j).

Proof. We puta =b = 0, A = Pn

i=1p_ix_i andB =Pn

j=1q_jy_j in TheoremA. Let X ={a_i}andY ={b_j}be independent discrete random variables with distributions P(X =x_i) = _P^pⁱ

n andP(Y = y_j) = _Q^qⁱ

n, 1 ≤ i ≤ n, respectively. So we have the desired inequalities.

Specially, if p_i = q_j = 1 (i, j = 1, . . . , n) in Theorem 4.1, then we have the following:

Corollary 4.2. Suppose thatx= (x₁, . . . , x_n)andy= (y₁, . . . , y_n)aren-tuples of nonnegative real numbers forn ≥ 2withPn

i=1x_i ∈ [0, c]andPn

i=1y_i ∈ [0, d]. If φ: [0, c]×[0, d]→Ris a concave function, then

(4.2) φ

n

X

i=1

xi,0

!

+φ 0,

n

X

j=1

yj

!

+ (n−2)φ(0,0)≤ 1 n

n

X

i=1 n

X

j=1

φ(xi, yj).

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5. Giaccardi’s Inequality

In 1955, Giaccardi (cf. [5, p.11]) proved the following inequality for a convex functionf : [a, A]→R,

n

X

i=1

p_if(x_i)≤C·f

n

X

i=1

p_ix_i

!

+D·(P_n−1)·f(x₀), where

C = Pn

i=1p_i(x_i−x₀) Pn

i=1p_ix_i−x₀ and D=

Pn i=1p_ix_i Pn

i=1p_ix_i−x₀ for a nonnegativen-tuplep = (p₁, . . . , p_n)withP_n := Pn

i=1p_i and a real(n+ 1)- tuplex= (x₀, x₁, . . . , x_n)such that fork = 0,1, . . . , n

a≤x_i ≤A, (x_k−x₀)

n

X

i=1

p_ix_i−x₀

!

≥0,

a <

n

X

i=1

p_ix_i < A and

n

X

i=1

p_ix_i 6=x₀.

In this section, we discuss a generalization of Giaccardi’s inequality to a function of two variables under similar conditions. Let x = (x₀, x₁, . . . , x_n) and y = (y₀, y₁, . . . , y_n) be nonnegative (n+ 1)-tuples, and p = (p₁, p₂, . . . , p_n) and q = (q₁, q₂, . . . , q_n)be nonnegativen-tuples with

(5.1) x₀ ≤x_k ≤

n

X

i=1

p_ix_i and y₀ ≤y_k ≤

n

X

j=1

q_jy_j fork = 1, . . . , n.

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We use the following notations:

P_n :=

n

X

i=1

p_i (≥0), Q_n:=

n

X

j=1

q_j (≥0),

K(X) :=

Pn

i=1p_ix_i−P_nx₀ Pn

i=1p_ix_i−x₀ , K(Y) :=

Pn

j=1q_jy_j−Q_ny₀ Pn

j=1q_jy_j−y₀ , L(X) := (Pn−1)Pn

i=1pixi

Pn

i=1p_ix_i−x₀ , L(Y) := (Q_n−1)Pn j=1q_jy_j Pn

j=1q_jy_j −y₀ ,

M(X, Y) :=

(

(PnQn−Pn−Qn)

n

X

i=1

pixi n

X

j=1

qjyj

+Q_ny₀

n

X

i=1

p_ix_i+P_nx₀

n

X

j=1

q_jy_j−P_nQ_nx₀y₀ )

× 1

(Pn

i=1p_ix_i−x₀) Pn

j=1q_jy_j −y₀ and

N(X, Y)

:=

(P_n−Q_n)

n

P

i=1

p_ix_i

n

P

j=1

q_jy_j −(P_n−1)Q_n

n

P

i=1

p_ix_iy₀+P_n(Q_n−1)x₀

n

P

j=1

q_jy_j _n

P

i=1

p_ix_i −x₀ _n

P

j=1

q_jy_j−y₀

! .

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Then we have the following theorem:

Theorem 5.1. Letφ: [x₀,Pn

i=1p_ix_i]×[y₀,Pn

j=1q_jy_j]→Rbe a concave function.

a)If

φ(x₀, y₀) +φ

n

X

i=1

p_ix_i,

n

X

j=1

q_jy_j

!

≥φ x₀,

n

X

j=1

q_jy_j

! +φ

n

X

i=1

p_ix_i, y₀

! , then

max (

Q_nK(X)φ

n

X

i=1

p_ix_i, y₀

!

+P_nK(Y)φ x₀,

n

X

j=1

q_jy_j

!

+M(X, Y)φ(x₀, y₀),

P_nL(Y)φ

n

X

i=1

p_ix_i, y₀

!

+Q_nL(X)φ x₀,

n

X

j=1

q_jy_j

!

−M(X, Y)φ

n

X

i=1

pixi,

n

X

j=1

qjyj

!)

≤

n

X

i=1 n

X

j=1

piqjφ(xi, yj).

b)If

φ(x₀, y₀) +φ

n

X

i=1

p_ix_i,

n

X

j=1

q_jy_j

!

≤φ x₀,

n

X

j=1

q_jy_j

! +φ

n

X

i=1

p_ix_i, y₀

! ,

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JJ II

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then

max (

Q_nK(X)φ

n

X

i=1

p_ix_i,

n

X

j=1

q_jy_j

!

+P_nL(Y)φ(x₀, y₀) +N(X, Y)φ x₀,

n

X

j=1

q_jy_j

! ,

P_nK(Y)φ

n

X

i=1

p_ix_i,

n

X

j=1

q_jy_j

!

+Q_nL(X)φ(x₀, y₀)

−N(X, Y)φ

n

X

i=1

pixi, y0

!)

≤

n

X

i=1 n

X

j=1

piqjφ(xi, yj).

Proof. LetX andY be as they were in the proof of Theorem 4.1, and puta = x₀, A=Pn

i=1p_ix_i,b =y₀ andB =Pn

j=1q_jy_j, and use TheoremA. Then we have the desired inequalities of this theorem.

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vol. 8, iss. 3, art. 88, 2007

Title Page Contents

JJ II

J I

Close

References

[1] V. CSISZÁRAND 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. IZUMINOAND M. TOMINAGA, Estimations in Hölder’s type inequalities, Math. Inequal. Appl., 4 (2001), 163–187.

[5] D. MITRINOVI ´C, J.E. PE ˇCARI ´CANDA.M. FINK, Classical and New Inequal- ities in Analysis, Kluwer. Acad. Pub., Boston, London 1993.

[6] J.E. PE ˇCARI ´C, F. PROSCHANANDY.L. TONG, Convex Functions, Partial Or- derings, and Statistical Applications, Mathematics in Science and Engineering, Georgia Institute of Technology, 1992.