Received27September,2006;accepted21April,2007CommunicatedbyI.Pinelis ∆ φ =∆ φ ( a,b,A,B ):= φ ( a,b ) − φ ( a,B ) − φ ( A,b )+ φ ( A,B ) . P .Forafunction φ : D → R ,weput [ b,B ] ( 0 ≤ a<A and 0 ≤ b<B ).Let E [ X ] betheexpectationofarandomvariable X wit

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

BOŽIDAR IVANKOVI ´C, SAICHI IZUMINO, JOSIP E. PE ˇCARI ´C, AND MASARU TOMINAGA FACULTY OFTRANSPORT ANDTRAFFICENGINEERING

UNIVERSITY OFZAGREB, VUKELI ´CEVA4 10000 ZAGREB, CROATIA

ivankovb@fpz.hr FACULTY OFEDUCATION

TOYAMAUNIVERSITY

GOFUKU, TOYAMA930-8555, JAPAN s-izumino@h5.dion.ne.jp FACULTY OFTEXTILETECHNOLOGY

UNIVERSITY OFZAGREB, PIEROTTIJEVA6 10000 ZAGREB, CROATIA

pecaric@mahazu.hazu.hr

TOYAMANATIONALCOLLEGE OFTECHNOLOGY

13, HONGO-MACHI, TOYAMA-SHI

939-8630, JAPAN

mtommy@toyama-nct.ac.jp

Received 27 September, 2006; accepted 21 April, 2007 Communicated by I. Pinelis

ABSTRACT. V. Csiszár and T.F. Móri gave an extension of Diaz-Metcalf’s inequality for con- cave 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.

Key words and phrases: Diaz-Metcalf inequality, Hölder’s inequality, Hadamard’s inequality, Petrovi´c’s inequality, Giac- cardi’s inequality.

2000 Mathematics Subject Classification. 26D15.

1. INTRODUCTION

In this paper, let(X, Y)be a random vector withP[(X, Y)∈ D] = 1whereD := [a, A]× [b, B](0 ≤a < Aand0≤ b < B). LetE[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).

127-07

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.

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 Theorem A can 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 Theorem A.

b)If∆φ ≤0,then

(1.2) max

k=3,4{λ_kE[X] +µkE[Y] +νk} ≤E[φ(X, Y)](≤φ(E[X], E[Y]), whereλ_k, µ_kandν_k(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 in- equalities in Theorem A represent reverse inequalities of it.

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

2. REVERSEHÖLDER’S INEQUALITY

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

∆φ=a^1pb^1q −a^1pB^1q −A^1pb^1q +A^p1B^1q =

A^1p −a^1p B^1q −b^1q

(>0).

Moreover, puttingA = B = 1, and replacingX, Y,a andbbyX^p, Y^q, α^p andβ^q, respec- tively, in Theorem A, we have the following result:

Theorem 2.1. Let p, q > 1 be real numbers with ¹_p + ¹_q = 1. Let 0 < α ≤ X ≤ 1 and 0< 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 Theorem 2.1 we have the following inequality related to Hölder’s inequality:

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

p^1pq^1q(β−αβ^q)^p1(α−α^pβ)^1qE[X^p]^p1E[Y^q]^1q (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]}^1p{q(α−α^pβ)E[Y^q]}^1q

=p^1pq^1q(β−αβ^q)^1p(α−α^pβ)^1qE[X^p]^1pE[Y^q]^1q. 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 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]

≤ (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 ≤ 1and0 < β ≤ Y ≤1, then

(2.6) E[X^p]^1pE[Y^q]^1q ≤ 1−α^pβ^q

p^1pq^1q(β−αβ^q)^1p(α−α^pβ)^1q

E[XY].

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 = {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 by Pn

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 inequalities

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^p1q^1q(β−αβ^q)^1p(α−α^pβ)^1q

n

X

i=1 n

X

j=1

w_iz_ja_ib_j.

3. HADAMARD’SINEQUALITY

The following well-known inequality is due to Hadamard [5, p.11]: For a concave function f : [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. LetXandY 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 Theorem A we have

λ₁E[X] +µ₁E[Y] +ν₁ =λ₂E[X] +µ₂E[Y] +ν₂ = φ(A, b) +φ(a, B) 2

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 The- orem 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. 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)whereu: [a, A]→Ris an integrable function withu(s) =u(a+A−s),RA

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

b v(t)dt = 1,v(t) =v(b+B−t). If φis 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

.

4. PETROVI ´C’SINEQUALITY

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)and p = (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 num- bers and putP_n := Pn

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

j=1q_j (> 0). Suppose that x= (x₁, . . . , x_n) andy = (y₁, . . . , y_n)aren-tuples of nonnegative real numbers with0 ≤ x_k ≤ Pn

i=1p_ix_i ≤ c and0 ≤ y_k ≤ Pn

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

a) Suppose

φ(0,0) +φ

n

X

i=1

pixi,

n

X

j=1

qjyj

!

≥φ

n

X

i=1

pixi,0

!

+φ 0,

n

X

j=1

qjyj

! .

a−(i) If _P¹

n + _Q¹

n ≤1, then (4.1) 1

Pn

φ

n

X

i=1

p_ix_i,0

! + 1

Qn

φ 0,

n

X

j=1

q_jy_j

! +

1− 1

Pn

− 1 Qn

φ(0,0)

≤ 1 P_nQ_n

n

X

i=1 n

X

j=1

p_iq_jφ(x_i, y_j).

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) IfP_n ≥Q_n, then 1

Pn

φ

n

X

i=1

p_ix_i,

n

X

j=1

q_jy_j

! +

1 Qn

− 1 Pn

φ 0,

n

X

j=1

q_jy_j

! +

1− 1

Qn

φ(0,0)

≤ 1 P_nQ_n

n

X

i=1 n

X

j=1

p_iq_jφ(x_i, y_j).

b−(iv) IfQ_n ≥P_n, then 1

Q_nφ

n

X

i=1

pixi,

n

X

j=1

qjyj

!

− 1

Q_n − 1 P_n

φ

n

X

i=1

pixi,0

! +

1− 1

P_n

φ(0,0)

≤ 1 P_nQ_n

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 Theorem A. LetX = {a_i} andY = {b_j} be independent discrete random variables with distributionsP(X = x_i) = _P^pi

n

andP(Y =y_j) = _Q^qi

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

Specially, ifp_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]. Ifφ : [0, c]×[0, d]→ Ris a concave function, then

(4.2) φ

n

X

i=1

x_i,0

!

+φ 0,

n

X

j=1

y_j

!

+ (n−2)φ(0,0)≤ 1 n

n

X

i=1 n

X

j=1

φ(x_i, y_j).

5. GIACCARDI’SINEQUALITY

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

n

X

i=1

pif(xi)≤C·f

n

X

i=1

pixi

!

+D·(Pn−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. Letx = (x₀, x₁, . . . , x_n)and y = (y₀, y₁, . . . , y_n)be non- negative (n + 1)-tuples, and p = (p1, p2, . . . , pn) and q = (q1, q2, . . . , qn) be nonnegative n-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.

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) := (P_n−1)Pn

i=1p_ix_i 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) :=

(

(P_nQ_n−P_n−Q_n)

n

X

i=1

p_ix_i

n

X

j=1

q_jy_j

+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−x_{0 n}

P

j=1

q_jy_j−y₀

! .

Then we have the following theorem:

Theorem 5.1. Letφ: [x0,Pn

i=1pixi]×[y0,Pn

j=1qjyj]→Rbe a concave function.

a)If

φ(x0, y0) +φ

n

X

i=1

pixi,

n

X

j=1

qjyj

!

≥φ x0,

n

X

j=1

qjyj

! +φ

n

X

i=1

pixi, 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

p_ix_i,

n

X

j=1

q_jy_j

!)

≤

n

X

i=1 n

X

j=1

p_iq_jφ(x_i, y_j).

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

! , then

max (

QnK(X)φ

n

X

i=1

pixi,

n

X

j=1

qjyj

!

+PnL(Y)φ(x0, y0) +N(X, Y)φ x0,

n

X

j=1

qjyj

! ,

PnK(Y)φ

n

X

i=1

pixi,

n

X

j=1

qjyj

!

+QnL(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. Let X and Y be as they were in the proof of Theorem 4.1, and put a = x₀, A = Pn

i=1pixi,b =y0andB =Pn

j=1qjyj, and use Theorem A. Then we have the desired inequal-

ities of this theorem.

REFERENCES

[1] V. CSISZÁRANDT.F. MÓRI, The convexity method of proving moment-type inequalities, Statist.

Probab. Lett., 66 (2004), 303–313.

[2] J.B. DIAZANDF.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. IZUMINOANDM. TOMINAGA, Estimations in Hölder’s type inequalities, Math. Inequal. Appl., 4 (2001), 163–187.

[5] D. MITRINOVI ´C, J.E. PE ˇCARI ´C AND A.M. FINK, Classical and New Inequalities in Analysis, Kluwer. Acad. Pub., Boston, London 1993.

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