Inequalities for a Weighted Multiple Integral

FENG QI

Abstract. In the article, using Taylor’s formula for functions of several vari-ables, the author establishes some inequalities for the weighted multiple inte-gral of a function defined on anm-dimensional rectangle, if its partial deriva-tives of (n+ 1)-th order remain between bounds. From which Iyengar’s in-equality is generalized and related results in references could be deduced.

1. Main Results

For given points a = (a1,_{· · ·} , am), b = (b1,_{· · ·} , bm) _{∈ R}m and ai < bi, i = 1,2,_{· · ·} , m, denote them-rectangles by

Qm=

m

Y

i=1

[ai, bi], Qm(t) =

m

Y

i=1

ai, ci(t),

(1)

whereci(t) = (1₋t)ai+tbi,i= 1,2,_{· · ·}, m,t_∈[0,1].

Let ν = (ν1,_{· · ·} , νm) be a multi-index, that is, νi = integer _> 0, with _|ν_| =

m

P

i=1

νi. Let f be a function of several variables defined on Qm, and its partial

derivatives of (n+1)-th order remain between the upper and lower boundsMn+1(ν) andNn+1(ν) as follows

Nn+1(ν)₆Dνf(x)₆Mn+1(ν), x_∈Qm,

(2)

where we define

Dνf(x) =∂n+1f(x) _Ym

i=1 ∂xνi

i .

(3)

Let w(x) _> 0 be an integrable function of several variables defined on the m -rectangleQm, which is not identically zero forx_∈Qm. Define

hs,ν(t) =

Z

Qm(t) w(x)

m

Y

i=1

(xi₋si)νidx,

(4)

wheres= (s1, s2,· · ·, sm)∈Rm,t∈[0,1].

In this article, using Taylor’s formula for functions with several variables, we obtain some inequalities for a weighted multiple integral R_Qmw(x)f(x)dx with weight w(x) _> 0 on the m-rectangle Qm in terms of the values of the partial

Date: November 29, 1999.

1991Mathematics Subject Classification. Primary 26D15; Secondary 26D10.

Key words and phrases. multiple integral, weighted integral, inequality, Taylor’s formula for functions of several variables,m-rectangle, bounded derivative.

The author was partially supported by NSF of the Education Committee of Henan Province, The People’s Republic of China.

derivatives of the functionfat pointsaandband the boundsMn+1(ν) andNn+1(ν)

ofDν_{f(x), that is}

Main Theorem. Let f _∈ Cn+1(Qm) and Nn+1(ν) ₆ Dνf(x) ₆ Mn+1(ν) hold for any x _∈ Qm and |ν| = n+ 1, where Mn+1(ν) and Nn+1(ν) are constants

depending on nandν. Letw(x)be an integrable function of several variables over

Qm, which is not identically zero. Then, for any t∈(0,1),

(i) if nis an even, we have

X

|ν|=n+1

Mn+1(ν)hb,ν(1)−Nn+1(ν)hb,ν(t)

m

Q

i=1

(νi!)

+ X

|ν|=n+1

Nn+1(ν) m

Q

i=1

(νi!)

ha,ν(t)

6

Z

Qm

w(x)f(x)dx₋ n

X

k=0

X

|ν|=k

Dν_f(b)

m

Q

i=1

(νi!)

[hb,ν(1)−hb,ν(t)]−

n

X

k=0

X

|ν|=k

Dν_f(a)

m

Q

i=1

(νi!) ha,ν(t)

6 X |ν|=n+1

Nn+1(ν)hb,ν(1)−Mn+1(ν)hb,ν(t)

m

Q

i=1

(νi!)

+ X

|ν|=n+1

Mn+1(ν)

m

Q

i=1

(νi!)

ha,ν(t);

(5)

(ii) if nis an odd,

X

|ν|=n+1

Nn+1(ν)hb,ν(1)−Mn+1(ν)hb,ν(t)

m

Q

i=1

(νi!)

+ X

|ν|=n+1

Nn+1(ν)

m

Q

i=1

(νi!)

ha,ν(t)

6

Z

Qm

w(x)f(x)dx₋ n

X

k=0

X

|ν|=k

Dνf(b)

m

Q

i=1

(νi!)

[hb,ν(1)−hb,ν(t)]−

n

X

k=0

X

|ν|=k

Dνf(a)

m

Q

i=1

(νi!) ha,ν(t)

6 X |ν|=n+1

Mn+1(ν)hb,ν(1)−Nn+1(ν)hb,ν(t)

m

Q

i=1

(νi!)

+ X

|ν|=n+1

Mn+1(ν) m

Q

i=1

(νi!)

ha,ν(t).

(6)

2. Proof of Main Theorem

Lett_∈(0,1) be a parameter, and write Z

Qm

w(x)f(x)dx= Z

Qm(t)

w(x)f(x)dx+ Z

Qm\Qm(t)

w(x)f(x)dx.

(7)

The well-known Taylor’s formula for a multivariable function states that

f(x) =

n X k=0 1 k! m X i=1

(xi₋ai) ∂

∂xi

!k

f(a) +Rn(x),

(8)

f(x) =

n X k=0 1 k! m X i=1

(xi₋bi) ∂

∂xi

!k

f(b) +rn(x),

where

Rn(x) = 1 (n+ 1)!

m

X

i=1

(xi₋ai) ∂

∂xi

!n+1

f(a+θ(x₋a)), θ_∈(0,1),

(10)

rn(x) =

1 (n+ 1)!

m

X

i=1

(xi−bi) ∂ ∂xi

!n+1

f(b+µ(x₋b)), µ_∈(0,1).

(11) Since m X i=1 qi !k

=k! X

|ν|=k m

Y

i=1 qνi

i νi!, (12)

integrating on both sides of (8) overQm(t) gives us

Z

Qm(t)

w(x)f(x)dx

= n X k=0 1 k! Z

Qm(t) w(x)

m

X

i=1

(xi₋ai) ∂

∂xi

!k

f(a)dx+ Z

Qm(t)

w(x)Rn(x)dx

=

n

X

k=0

X

|ν|=k

1

m

Q

i=1

(νi!)

Z

Qm(t) w(x)

m

Y

i=1

(xi₋ai) ∂

∂xi

νi

f(a)dx

+ X

|ν|=n+1

1

m

Q

i=1

(νi!)

Z

Qm(t) w(x)

m

Y

i=1

(xi₋ai) ∂

∂xi

νi

f(a+θ(x₋a))dx

(13) = n X k=0 X

|ν|=k

1

m

Q

i=1

(νi!)

∂kf(a)

m

Q

i=1 ∂xνi_i

Z

Qm(t) w(x)

m

Y

i=1

(xi₋ai)νidx

+ X

|ν|=n+1

1

m

Q

i=1

(νi!) Z

Qm(t) w(x)

m

Y

i=1

(xi₋ai)νi

·∂

n+1_f(a+θ(x

−a))

m

Q

i=1 ∂xνi_i

dx = n X k=0 X

|ν|=k

Dν_f(a)

m

Q

i=1

(νi!)

ha,ν(t)

+ X

|ν|=n+1

1

m

Q

i=1

(νi!) Z

Qm(t) w(x)

m

Y

i=1

Using inequality (2) and computing directly yields

X

|ν|=n+1

Nn+1(ν)

m

Q

i=1

(νi!) ha,ν(t)

≤ X

|ν|=n+1

1

m

Q

i=1

(νi!)

Z

Qm(t) w(x)

m

Y

i=1

(xi₋ai)νi_Dν_{f(a+θ(x−a))dx}

(14)

≤ X

|ν|=n+1

Mn+1(ν) m

Q

i=1

(νi!)

ha,ν(t).

The combination of (13) and (14) leads to

X

|ν|=n+1

Nn+1(ν)

m

Q

i=1

(νi!) ha,ν(t)

≤ Z

Qm(t)

w(x)f(x)dx₋ n

X

k=0

X

|ν|=k

Dνf(a)

m

Q

i=1

(νi!) ha,ν(t)

(15)

≤ X

|ν|=n+1

Mn+1(ν)

m

Q

i=1

(νi!)

Integrating (9) on the domainQm\Qm(t), we arrive at

Z

Qm\Qm(t)

w(x)f(x)dx

= n X k=0 1 k! Z Qm\Qm(t) w(x)

m

X

i=1

(xi−bi) ∂ ∂xi

!k

f(b)dx+ Z

Qm\Qm(t)

rn(x)dx

=

n

X

k=0

X

|ν|=k

1

m

Q

i=1

(νi!) Z

Qm w(x)

m

Y

i=1

(xi−bi) ∂ ∂xi

νi f(b)dx

−

n

X

k=0

X

|ν|=k

1

m

Q

i=1

(νi!) Z

Qm(t) w(x)

m

Y

i=1

(xi₋bi) ∂

∂xi

νi f(b)dx

+ X

|ν|=n+1

1

m

Q

i=1

(νi!)

Z

Qm w(t)

m

Y

i=1

(xi₋bi) ∂

∂xi

νi

f(b+µ(x₋b))dx

(16)

− X

|ν|=n+1

1

m

Q

i=1

(νi!)

Z

Qm(t) w(t)

m

Y

i=1

(xi₋bi) ∂

∂xi

νi

f(b+µ(x₋b))dx

=

n

X

k=0

X

|ν|=k

Dνf(b)

m

Q

i=1

(νi!)

[hb,ν(1)−hb,ν(t)]

+ X

|ν|=n+1

1

m

Q

i=1

(νi!)

Z

Qm w(t)

m

Y

i=1

(xi−bi) ∂ ∂xi

νi

f(b+µ(x₋b))dx

− X

|ν|=n+1

1

m

Q

i=1

(νi!)

Z

Qm(t) w(t)

m

Y

i=1

(xi−bi) ∂ ∂xi

νi

f(b+µ(x₋b))dx.

Similar to the deduction of (14), ifnis an odd, we have X

|ν|=n+1

Nn+1(ν)

m

Q

i=1

(νi!) hb,ν(t)

6 X |ν|=n+1

1

m

Q

i=1

(νi!)

Z

Qm(t) w(t)

m

Y

i=1

(xi₋bi) ∂

∂xi

νi

f(b+µ(x₋b))dx

(17)

= X

|ν|=n+1

1

m

Q

i=1

(νi!) Z

Qm(t) w(x)

m

Y

i=1

(xi₋bi)νi_Dν_f(b+µ(x

−b))dx

6 X |ν|=n+1

Mn+1(ν)

m

Q

i=1

(νi!)

hb,ν(t);

Substituting (17) into (16) we have that, ifnis an odd number, then X

|ν|=n+1

Nn+1(ν)hb,ν(1)−Mn+1(ν)hb,ν(t)

m

Q

i=1

(νi!)

≤ Z

Qm\Qm(t)

w(x)f(x)dx₋ n

X

k=0

X

|ν|=k

Dν_f(b)

m

Q

i=1

(νi!)

[hb,ν(1)−hb,ν(t)]

(18)

≤ X

|ν|=n+1

Mn+1(ν)hb,ν(1)−Nn+1(ν)hb,ν(t)

m

Q

i=1

(νi!)

;

ifnis an even number, then the inequalities in (18) are reversed.

By addition of inequalities (15) and (18), the Main Theorem was proved.

Remark 1. It is noted that we also can consider the similar estimates for the weighted multiple integral R_Qmw(x)f(x)dx on the m-dimensional ball centered atawith radius_|b₋a_|, that is,Qm=Ba(_|b₋a_|),a, b_∈Rm_.

Remark 2. In the Main Theorem, if we take m = 1, we can obtain the results in [14]; if we set m = 1 and w(x) = 1, then we get the results in [12]; if we let

w(x) = 1, we have the results in [13]. In particular, if we takew(x) = 1,m= 1 and

n= 0, the Iyengar inequality [6] is deduced, which has been generalized by many mathematicians in [1, 2, 3, 4, 5, 8, 11, 15] (also see [7, 9, 10]).

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Department of Mathematics, Jiaozuo Institute of Technology, Jiaozuo City, Henan 454000, The People’s Republic of China