New Grüss type inequalities for double integrals

Applied Mathematics and Computation 228 (2014) 102–107

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

New Grüss type inequalities for double integrals Mohammad W. Alomari Department of Mathematics, Faculty of Science and Information Technology, Jadara University, 21110 Irbid, Jordan

a r t i c l e

i n f o

a b s t r a c t In this paper, new Grüss type inequalities for double integrals are proved. Some sharp bounds are provided as well. Ó 2013 Elsevier Inc. All rights reserved.

Keywords: ˇ ebyšev functional C Grüss inequality Function of bounded variation

1. Introduction ˇ ebyšev functional For a; b; c; d 2 R, we consider the subset D :¼ fðx; yÞ : a 6 x 6 b; c 6 y 6 dg # R2 . The C

T ðf ;gÞ :¼

1 ðb  aÞðd  cÞ

Z

b

Z

a

d

f ðt; sÞgðt; sÞdsdt  c

1 ðb  aÞðd  cÞ

Z a

b

Z

d

f ðt; sÞdsdt

c

1 ðb  aÞðd  cÞ

Z a

b

Z

d

gðt;sÞdtds

ð1:1Þ

c

has interesting applications in the approximation of the integral of a product as pointed out in the references below. In 2001, Hanna et al. [14] have proved the following inequality which is of Grüss type for double integrals, where f and g satisfy that

jf ðx; yÞ  f ðu; v Þj 6 M1 jx  uja1 þ M2 jy  v ja2 ; where, M 1 ; M 2 > 0; a1 ; a2 2 ð0; 1,

jgðx; yÞ  gðu; v Þj 6 N1 jx  ujb1 þ N2 jy  v jb2 ; where, N 1 ; N 2 > 0; b1 ; b2 2 ð0; 1. Then, a þb

jT ðf ; gÞj 6 M 1 N1

a

þ M2 N1

a

b

ðb  aÞ 1 1 2ðb  aÞ 1 ðd  cÞ 2 þ M1 N2 ða1 þ b1 þ 1Þða1 þ b1 þ 2Þ ða1 þ 1Þða1 þ 2Þðb2 þ 1Þðb2 þ 2Þ b

a þb

2ðb  aÞ 2 ðd  cÞ 1 ðb  aÞ 2 2 : þ M2 N2 ða2 þ 1Þða2 þ 2Þðb1 þ 1Þðb1 þ 2Þ ða2 þ b2 þ 1Þða2 þ b2 þ 2Þ

ð1:2Þ

When a1 ¼ a2 ¼ 1 and b1 ¼ b2 ¼ 1, then

jf ðx; yÞ  f ðu; v Þj 6 L1 jx  uj þ L2 jy  v j; jf ðx; yÞ  f ðu; v Þj 6 K 1 jx  uj þ K 2 jy  v j; where, L1 ; L2 ; K 1 ; K 2 > 0 then (1.2) becomes 2

jT ðf ; gÞj 6 L1 K 1

2

ðb  aÞ ðb  aÞðd  cÞ ðb  aÞðd  cÞ ðd  cÞ þ L1 K 2 þ L2 K 1 þ L2 K 2 : 18 18 12 12

E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.11.093

ð1:3Þ

103

M.W. Alomari / Applied Mathematics and Computation 228 (2014) 102–107

After that, in 2002, Pachpatte [15] has established two inequalities of Grüss type involving continuous functions of two independent variables whose first and second partial derivatives are exist, continuous and belong to L1 ðDÞ; for details see [15]. For more recent results about multivariate and multidimensional Grüss type inequalities the reader may refer to [1–3,16], for another Grüss type inequalities see [5,6] and [8–12]. Functions of bounded variation are of great interest and usefulness because of their valuable properties and multiple applications in several subfields including rectifiable curves, Fourier series, Stieltjes integrals, the calculus of variations and others. According to Clarkson and Adams [7], several definitions have been given under which a function of two or more independent variables shall be said to be of bounded variation. Among of these definitions, six are usually associated with the names of Vitali, Hardy, Arzelà, Pierpont, Fréchet, and Tonelli. For more details about these definitions, the interested reader may refer to [4,7] and the recent book [13]. In this paper, we are mainly interested with the Arzelà definition, as follows: Let P :¼ fðxi ; yi Þ : xi1 6 x 6 xi ; yi1 6 y 6 yi ; i ¼ 1; . . . ; ng be a partition of D, write

Df ðxi ; yi Þ ¼ f ðxi ; yi Þ  f ðxi1 ; yi1 Þ for i ¼ 1; 2; . . . ; n. The function f ðx; yÞ is said to be of bounded variation in the Arzelà sense (or simply bounded variation) if P there exists a positive quantity M such that for every partition on D we have ni¼1 jDf ðxi ; yi Þj 6 M. Therefore, one can define the concept of total variation of a function of two variables, as follows: P P Let f be of bounded variation on D, and let ðPÞ denote the sum ni¼1 jDf ðxi ; yi Þj corresponding to the partition P of D. The number

nX o _ ðf Þ :¼ sup P : P 2 PðDÞ D

is called the total variation of f on D. ˇ ebyšev function (1.1) under various assumptions. The aim of this paper is to establish several new bounds for the C 2. The results We may start with the following result: Theorem 1. Let f ; g : D ! R be such that f satisfies

jf ðx; yÞ  f ðu; v Þj 6 M1 jx  uja1 þ M 2 jy  v ja2 ; where, M 1 ; M 2 > 0; a1 ; a2 2 ð0; 1 and there exists the real numbers c; C such that c 6 gðx; yÞ 6 C for all ðx; yÞ 2 D, then

jT ðf ; gÞj 6

1 a a ðC  cÞ½M 1 ðb  aÞ 1 þ M2 ðd  cÞ 2 : 4

ð2:1Þ

Proof. We use the notation D :¼ ðb  aÞðd  cÞ. By simple calculations, we obtain the identity

1 T ðf ; gÞ :¼ D

Z a

b

Z c

d



#  " Z Z f ða; cÞ þ f ða; dÞ þ f ðb; cÞ þ f ðb; dÞ 1 b d  gðx; yÞ  f ðx; yÞ  gðt; sÞdtds dxdy: D a c 4

ð2:2Þ

Taking the modulus in (2.2) and utilizing the triangle inequality, we get

 Z Z  #  " Z Z  1  b d f ða; cÞ þ f ða; dÞ þ f ðb; cÞ þ f ðb; dÞ 1 b d   gðx; yÞ  f ðx; yÞ  gðt; sÞdtds dxdy   4 D a c D a c    Z b Z d Z bZ d    1  f ðx; yÞ  f ða; cÞ þ f ða; dÞ þ f ðb; cÞ þ f ðb; dÞ  gðx; yÞ  1 6 gðt; sÞdtdsdxdy    4 D a c  D a c    Z b Z d  Z bZ d    1 f ða; cÞ þ f ða; dÞ þ f ðb; cÞ þ f ðb; dÞ 1    gðt; sÞdtdsdxdy: 6 sup f ðx; yÞ  gðx; yÞ     4 D a c D a c ðx;yÞ2D

jT ðf ; gÞj ¼

Now, we define

I :¼

1 D

Z c

then, we have

d

Z a

b

gðx; yÞ 

1 D

Z c

d

Z

b

!2 gðt; sÞdtds

a

dxdy;

ð2:3Þ

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M.W. Alomari / Applied Mathematics and Computation 228 (2014) 102–107

Z

1 I :¼ D

Z

c

Z

1 D

¼

d

b

Z

c

Z

4g ðx; yÞ  2gðx; yÞ 1 D 2

a

d

2

b

Z

1 D

g 2 ðx; yÞdxdy 

a

Z

d

c

d

a

Z

c

b

1 gðt; sÞdtds þ D

Z

d

Z

c

b

a

!2 3 gðt; sÞdtds 5dxdy

!2

b

g ðt; sÞdtds

:

a

On the other hand, we have

I¼

1 D

C

Z

d

Z

c

!

b

1 D

gðt; sÞdtds a

Z c

d

Z

!

b

gðt; sÞdtds  c 

a

1 D

Z

d

Z

c

b

½C  gðt; sÞ  ½gðt; sÞ  cdtds:

a

As c 6 gðx; yÞ 6 C, for all ðx; yÞ 2 D, then

1 D

Z

Z

d

c

b

½C  gðt; sÞ  ½gðt; sÞ  cdtds P 0;

a

which implies that

1 C ðb  aÞðd  cÞ

I6

Z

d

c

Z

!

b

gðt; sÞdtds

a

1 D

Z

d

c

Z

!

b

gðt; sÞdtds  c :

a

Using the elementary inequality

ðC  AÞðB  CÞ 6

1 ðB  AÞ2 ; 4

which holds for all A; B; C 2 R, we get

I6

1 ðC  cÞ2 : 4

ð2:4Þ

Using Cauchy–Buniakowski–Schwarz’s integral inequality we have

"

1 IP D

Z c

d

Z

b

a

  #2 Z Z   1 d b   gðt; sÞdtdsdxdy ; gðx; yÞ    D c a

which gives by (2.4),

Z

d

Z

c

a

b

  Z Z   1 d b 1   gðt; sÞdtdsdxdy 6 ðC  cÞðb  aÞðd  cÞ: gðx; yÞ    2 D c a

ð2:5Þ

Now, by assumptions we have

    f ðx; yÞ  f ða; cÞ þ f ða; dÞ þ f ðb; cÞ þ f ðb; dÞ   4 1 6 ½jf ðx; yÞ  f ða; cÞ þ f ðx; yÞ  f ða; dÞ þ f ðx; yÞ  f ðb; cÞ þ f ðx; yÞ  f ðb; dÞj 4 1 6 ½jf ðx; yÞ  f ða; cÞj þ jf ðx; yÞ  f ða; dÞj þ jf ðx; yÞ  f ðb; cÞj þ jf ðx; yÞ  f ðb; dÞj 4 1 6 ½M 1 jx  aja1 þ M 2 jy  cja2 þ M1 jx  aja1 þ M2 jy  dja2 þ M 1 jx  bja1 þ M2 jy  cja2 þ M 1 jx  bja1 þ M2 jy  dja2  4 M1 M2 6 ½jx  aja1 þ jx  bja1  þ ½jy  cja2 þ jy  dja2 : 2 2 It follows that,

     f ða; cÞ þ f ða; dÞ þ f ðb; cÞ þ f ðb; dÞ M 1 M2 a  a  sup f ðx; yÞ  sup jy  cja2 þ jy  dj 2 jx  aja1 þ jx  bj 1 þ  6 2 sup 4 2 y x

ðx;yÞ2D

¼

1 a a  M1 ðb  aÞ 1 þ M 2 ðd  cÞ 2 : 2

Combining (2.5) and (2.6) with (2.3), we get the required result (2.1). Corollary 1. Let g be as in Theorem 1. If f : D ! R satisfies that

jf ðx; yÞ  f ðu; v Þj 6 M1 jx  uj þ M 2 jy  v j;

h

ð2:6Þ

M.W. Alomari / Applied Mathematics and Computation 228 (2014) 102–107

105

where, M 1 ; M 2 > 0, then

jT ðf ; g Þ j 6

1 ðC  cÞ½M 1 ðb  aÞ þ M2 ðd  cÞ: 4

ð2:7Þ

The following result holds: Theorem 2. Let f ; g : D ! R be such that f as in Theorem 1 and g is of bounded variation on D, then

 a a  ðb  a Þ 1 ðd  c Þ 2 _  ðg Þ: jT ðf ; g Þ j 6 M 1 þ M2 a1 þ 1 a2 þ 1 D

ð2:8Þ

Proof. As in Theorem 1, we observed that

   Z Z  Z Z   f ða; cÞ þ f ða; dÞ þ f ðb; cÞ þ f ðb; dÞ 1 b d  1 b d   g ðt; sÞdtdsdxdy: jT ðf ; g Þj 6 sup f ðx; yÞ  g ðx; yÞ  D   4 D ðx;yÞ2D a c a c However, since g is of bounded variation D, then

  Z Z Z Z   _ 1 b d 1 d b   sup g ðx; yÞ  g ðt; sÞdtds 6 sup jg ðx; yÞ  g ðt; sÞj  dtds 6 ðg Þ:   D D ðx;yÞ2D ðx;yÞ2D a c c a D Also, as we have

    f ðx; yÞ  f ða; cÞ þ f ða; dÞ þ f ðb; cÞ þ f ðb; dÞdxdy   4 a c Z bZ d 1 ½jf ðx; yÞ  f ða; cÞj þ jf ðx; yÞ  f ða; dÞjþjf ðx; yÞ  f ðb; cÞj þ jf ðx; yÞ  f ðb; dÞjdxdy 6 4D a c Z b Z d a a   M1 M2 ðb  aÞ 1 ðd  cÞ 2 a  a  6 jx  aja1 þ jx  bj 1 dx þ jy  cja2 þ jy  dj 2 dy 6 M 1 þ M2 : 2ðb  aÞ a 2ð d  c Þ c a1 þ 1 a2 þ 1

Z

1 D

b

Z

d

Combining the above inequalities we get the required result (2.8). h Corollary 2. Let g be as in Theorem 2. If f : D ! R satisfies that

jf ðx; yÞ  f ðu; v Þj 6 M1 jx  uj þ M 2 jy  v j; where, M 1 ; M 2 > 0, then

jT ðf ; g Þ j 6

_ 1 ½M 1 ðb  aÞ þ M 2 ðd  cÞ  ðg Þ: 2 D

ð2:9Þ

3. Sharp inequalities The following result holds: Theorem 3. Let f ; g : D ! R be such that f is of bounded variation on D and g be as in Theorem 1, then

jT ðf ; g Þ j 6 where,

W

_ 1 ðC  cÞ  ð f Þ; 8 D

ð f Þ is the total variation of f over D. The constant

D

ð3:1Þ 1 8

is the best possible.

Proof. As in Theorem 1, we observed that

   Z Z  Z Z   f ða; cÞ þ f ða; dÞ þ f ðb; cÞ þ f ðb; dÞ 1 b d  1 b d    g ðt; sÞdtdsdxdy: jT ðf ; g Þj 6 sup f ðx; yÞ  g ðx; yÞ    4 D a c  D a c ðx;yÞ2D However, since there exists c; C P 0 such that c 6 gðx; yÞ 6 C for all ðx; yÞ 2 D, then

1 D

Z c

d

Z a

b

  Z Z   1 d b 1   g ðt; sÞdtdsdxdy 6 ðC  cÞ: g ðx; yÞ    2 D c a

106

M.W. Alomari / Applied Mathematics and Computation 228 (2014) 102–107

Since, f is of bounded variation on D, we have that

   f ða; cÞ þ f ða; dÞ þ f ðb; cÞ þ f ðb; dÞ 1 sup f ðt; sÞ   6 4 sup ½jf ðt; sÞ  f ða; cÞj þ jf ðt; sÞ  f ða; dÞj þ jf ðt; sÞ 4 ðt;sÞ2D ðt;sÞ2D 1_ ð f Þ: f ðb; cÞj þ jf ðt; sÞ  f ðb; dÞj 6 4 D Combining the above obtained inequalities, we obtain the required result (3.1). To prove the sharpness of (3.1) holds with constant C > 0, i.e.,

jT ðf ; g Þj 6 C ðC  cÞ 

_

ðfÞ

ð3:2Þ

D

and consider the functions f ; g : D ! R be defined as

    aþb cþd  sgn s  : f ðt; sÞ ¼ g ðt; sÞ ¼ sgn t  2 2 Observe that f is of bounded variation on D and Rd Rb 0; c a f ðt; sÞg ðt; sÞdtds ¼ ðb  aÞðd  cÞ, making use of (3.2) we get the proof is completely finished. h

W

Rd Rb Rd Rb ð f Þ ¼ 4; C  c ¼ 2; c a f ðt; sÞdtds ¼ c a g ðt; sÞdtds ¼ 1 C, which proves that 8 is the best possible and thus

1D 6 8

Theorem 4. Let f ; g : D ! C be such that f is of bounded variation on D and g is a Lebesgue integrable function on D, then

1_ 1 ðfÞ  jT ðf ; g Þj 6 4 D D The constant

1 4

Z

b

Z

a

d

c

  Z Z   1 b d   g ðt; sÞdtdsdxdy: g ðx; yÞ    D a c

ð3:3Þ

is the best possible.

Proof. As in Theorem 1, we observed that

  Z  f ða; cÞ þ f ða; dÞ þ f ðb; cÞ þ f ðb; dÞ 1  jT ðf ; g Þj 6 sup f ðx; yÞ   D 4 ðx;yÞ2D

b

a

Z c

d

  Z Z   1 b d   g ðt; sÞdtdsdxdy: g ðx; yÞ    D a c

Since, f is of bounded variation on D, we have that

   f ða; cÞ þ f ða; dÞ þ f ðb; cÞ þ f ðb; dÞ 1 sup f ðx; yÞ   6 4 sup ½jf ðx; yÞ  f ða; cÞj þ jf ðx; yÞ 4

ðx;yÞ2D

ðx;yÞ2D

1_ ð f Þ: f ða; dÞj þ jf ðx; yÞ  f ðb; cÞj þ jf ðx; yÞ  f ðb; dÞj 6 4 D Combining the obtained inequalities, we obtain the required result (3.3). To prove the sharpness of (3.3) holds with constant C 1 > 0, i.e.,

_

1 jT ðf ; g Þj 6 C 1 ð f Þ  D D

Z a

b

Z c

d

  Z Z   1 b d   g ðt; sÞdtdsdxdy g ðx; yÞ    D a c

ð3:4Þ

and consider the functions f ; g : D ! R be defined as

    aþb cþd  sgn s  ; f ðt; sÞ ¼ sgn t  2 2 Observe that f is of bounded variation on D and 1 ðb 16

2

2

   aþb cþd g ðt; sÞ ¼ t  s : 2 2 W D

ð f Þ ¼ 4,

Rd Rb c

a

g ðt; sÞdtds ¼ 0;

Rd Rb c

a

jg ðt; sÞjdtds ¼

Rd Rb c

a

f ðt; sÞg ðt; sÞdtds ¼

 aÞ ðd  cÞ , making use of (3.4) we get 14 6 C 1 , which proves that 14 is the best possible and thus the proof is completely finished. h The variance of the function f : D ! C which is square integrable on D by Kð f Þ and is defined as:

2 2 31=2  Z Z Z Z  1 d b h i1=2 1 d b   2 4 Kð f Þ :¼ T f ; f ¼ f ðt; sÞdtds 5 ; jf ðt; sÞj dtds    D c a D c a where, D ¼ ðb  aÞðd  cÞ and f denotes the complex conjugate function of f. Corollary 3. Let f : D ! R be a mapping of bounded variation on D, then

ð3:5Þ

M.W. Alomari / Applied Mathematics and Computation 228 (2014) 102–107

1_ ð f Þ: 4 D

Kð f Þ 6 The constant

1 4

107

ð3:6Þ

is the best possible.

Proof. Applying Theorem 3 for g ¼ f we get

1_ 1 K ðfÞ 6 ðfÞ  4 D D 2

Z

Z

b

a

d

c

  Z Z   1 b d   f ðt; sÞdtdsdxdy: f ðx; yÞ    D a c

ð3:7Þ

By the Cauchy–Bunyakovsky–Schwarz integral inequality we have

1 D

Z

b

a

Z c

d

  Z Z   1 b d   f ðt; sÞdtdsdxdy 6 Kð f Þ; f ðx; yÞ    D a c

ð3:8Þ

  combining (3.7) and (3.8) we get the required result. Now, if we choose f : D ! R with f ðt; sÞ ¼ sgn t  aþb  sgn s  cþd , 2 2 then we obtain the sharpness of the constant 14. We shall omit the details. h Now we can state the following result when both functions are of bounded variation: Theorem 5. If f ; g : D ! C are of bounded variation on D, then

jT ðf ; g Þ j 6 The constant

1 16

_ 1 _ ð f Þ  ðg Þ: 16 D D

ð3:9Þ

is the best possible.

Proof. On making use of Theorem 3 and Corollary 3 we have

1_ 1 ðfÞ  jT ðf ; g Þ j 6 4 D D

Z a

b

Z

d c

  Z Z   _ 1 b d 1_ 1 _   g ðt; sÞdtdsdxdy 6 ð f Þ  Kðg Þ 6 ð f Þ  ðg Þ: g ðx; yÞ    4 D 16 D D a c D

The case of equality is obtained in (3.9) for

    aþb cþd  sgn s  ; f ðt; sÞ ¼ g ðt; sÞ ¼ sgn t  2 2 we shall omit the details.

h

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