A Generalisation Of Chebyshev’s Inequality For Functions Of Several Variables

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FUNCTIONS OF SEVERAL VARIABLES

P. CERONE, S.S. DRAGOMIR, G. HANNA, AND J.E. PE ˇCARI ´C

Abstract. The current note serves to develop generalisations of Chebyshev’s inequality for H¨older functions of several variables.

1. _Introduction

For two measurable functions f, g : [a, b] → R, define the functional, which is known in the literature as Chebychev’s functional

(1.1) T(f, g;a, b) := 1

b−a Z b

a

f(x)g(x)dx− 1

(b−a)2

Z b

a

f(x)dx· Z b

a

g(x)dx,

provided that the involved integrals exist. Results involving (1.1) abound in the literature, see for example [1] – [13].

The following inequality is well known as the Gr¨uss inequality [10]

(1.2) |T(f, g;a, b)| ≤ 1

4(M−m) (N−n),

provided thatm≤f ≤M andn≤g≤N a.e. on [a, b], wherem, M, n, N are real numbers. The constant 1

4 in (1.2) is the best possible.

Another inequality of this type is due to Chebychev (see for example [13, p. 207]). Namely, iff, g are absolutely continuous on [a, b] andf0, g0 ∈L∞[a, b] and

kf0k_∞:=ess sup

t∈[a,b]

|f0(t)|,then

(1.3) |T(f, g;a, b)| ≤ 1

12kf

0_k ∞kg

0_k

∞(b−a)

2

and the constant ₁₂1 is the best possible.

Cerone and Dragomir [2] have pointed out generalizations of the above results for integrals defined on two different intervals [a, b] and [c, d].

They defined the functional (generalised Chebychev functional)

T(f, g;a, b, c, d) : =M(f g;a, b) +M(f g;c, d) (1.4)

−M(f;a, b)M(g;c, d)−M(f;c, d)M(g;a, b),

where the integral mean is defined by

(1.5) M(f;a, b) := 1

b−a Z b

a

f(x)dx

Date: September 29, 2000.

1991Mathematics Subject Classification. Primary 26D15, 26D10; Secondary 26D99.

Key words and phrases. Chebychev inequality, H¨older, Chebychev functional, Several Variables.

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and obtained a variety of bounds using a generalisation of Korkine’s identity. Budimir, Cerone and Peˇcari´c [1] obtained the bounds for (1.4) for f and g of H¨older type and also, for a weighted version of (1.4). In particular, they obtained the following result.

Theorem 1. Let f, g : I ⊆ _R→_R be measurable on I and the intervals [a, b],

[c, d]⊂I. Further, suppose thatf and g are of H¨older type so that for x∈[a, b],

y∈[c, d]

(1.6) |f(x)−f(y)| ≤H1|x−y|r and |g(x)−g(y)| ≤H2|x−y|s,

whereH1,H2>0 andr, s∈(0,1]are fixed. The following inequality then holds,

(θ+ 1) (θ+ 2)|T(f, g;a, b, c, d)|

(1.7)

≤ H1H2

(b−a) (d−c)

h

|b−c|θ+2− |b−d|θ+2+|d−a|θ+2− |c−a|θ+2i,

whereθ=r+sandT(f, g;a, b, c, d)is as defined by (1.4) and (1.5).

Hanna, Dragomir and Cerone [11] obtained bounds for a two dimensional Cheby-chev functional for functions of H¨older type and applied them for perturbed Taylor-like formulae inR2.

It is the express aim of this note to obtain bounds for a Chebychev functional defined on ann−dimensional hypercube where the functions are of H¨older type.

2. Bounds for the Chebychev Functional

If we consider the Chebyshev functional:

Dn(f, g) : =

1

Qn

k=1(bk−ak)

Z b1

a1

· · · Z bn

an

f(x1, . . . , xn)g(x1, . . . , xn)dx1. . . dxn

−_Qn 1

k=1(bk−ak)

2

Z b1

a1

· · · Z bn

an

f(x1, . . . , xn)dx1. . . dxn

× Z b1

a1

· · · Z bn

an

g(x1, . . . , xn)dx1. . . dxn

= 1

ν

¯

a,¯b Z ¯b

¯

a

f(¯x)g(¯x)dx¯− 1

ν

¯

a,¯b2 Z ¯b

¯

a

f(¯x)dx¯· Z ¯b

¯

a

g(¯x)dx,¯

where ν

¯

a,¯b

:=Qn_k₌₁(bk−ak), then we can state the following generalisation

of Chebyshev’s inequality. Theorem 2. Let f, g:

¯

a,¯b

→_Rbe of H¨older type. That is,

|f(¯x)−f(¯y)| ≤ n X

i=1

Li|xi−yi|

pi_, _x,_¯ _y_¯_∈

¯

a,¯b ,

(2.1)

|g(¯x)−g(¯y)| ≤ n X

i=1

Hi|xi−yi| qi

, ¯x,y¯∈

¯

a,¯b ,

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whereLi,Hi>0andpi,qi∈(0,1]are fixed fori= 1,2, . . . , n.

Then we have the inequality:

|Dn(f, g)| ≤ n X

i=1

LiHi

(bi−ai) pi+qi

(pi+qi+ 1) (pi+qi+ 2)

(2.3)

+2

n X

i6=j i,j=1

LiHj

(bi−ai) pi₍_b

j−aj) qj

(pi+ 1) (pi+ 2) (qj+ 1) (qj+ 2)

and the inequality is sharp.

Proof. We have

(2.4) |f(¯x)−f(¯y)| ≤ n X

i=1

Li|xi−yi| pi

and

(2.5) |g(¯x)−g(¯y)| ≤ n X

i=1

Hi|xi−yi| qi_.

If we multiply (2.4) and (2.5), we may get

|(f(¯x)−f(¯y)) (g(¯x)−g(¯y))|

≤ n X

i,j=1

LiHj|xi−yi| pi_|_x

j−yj| qj

=

n X

i=1

LiHi|xi−yi| pi+qi₊

n X

i6=j i,j=1

LiHj|xi−yi| pi_|_x

j−yj| qj_.

If we integrate over ¯x,y¯ ∈

¯

a,¯b

= Qn_i₌₁[ai, bi] := [a1, b1]×[an, bn] we get from

Korkine’s identity

|Dn(f, g)|

(2.6)

≤ 1

2 [Qn_i₌₁(bi−ai)]

2

Z ¯b

¯

a Z ¯b

¯

a

|(f(¯x)−f(¯y)) (g(¯x)−g(¯y))|dxd¯ y¯

≤ 1

2

" n X

i=1

LiHi Z ¯b

¯

a Z ¯b

¯

a

|xi−yi|

pi+qi_d_xd_¯ _y_¯

+

n X

i6=j i,j=1

LiHj Z ¯b

¯

a Z ¯b

¯

a

|xi−yi| pi_|_x

j−yj| qj_d_xd_¯ _y_¯



 .

Now, we have that

Ai:= Z ¯b

¯

a Z ¯b

¯

a

|xi−yi|

pi+qi_d_xd_¯ _y_¯₌ n Y

k6=i k=1

(bk−ak)

2Z bi

ai Z bi

ai

|xi−yi| pi+qi_dx

idyi

and as

(2.7)

Z d

c Z d

c

|x−y|rdxdy= 2 (d−c)

r+2

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then we get

Ai =

n Y

k6=i k=1

(bk−ak)

2

· 2 (bi−ai) pi+qi+2

(pi+qi+ 1) (pi+qi+ 2)

= 2

n Y

k=1

(bk−ak)

2 (bi−ai)pi+qi

(pi+qi+ 1) (pi+qi+ 2) .

Also,

Aij : =

Z ¯b

¯

a Z ¯b

¯

a

|xi−yi| pi

|xj−yj| qj

dxd¯ y¯

=

n Y

k6=i,j k=1

(bk−ak)

2Z bi

ai Z bi

ai

|xi−yi| pi_dx

idyi· Z bj

aj Z bj

aj

|xj−yj| qj_dx

jdyj

=

n Y

k6=i,j k=1

(bk−ak)2·

2 (bi−ai)pi+2

(pi+ 1) (pi+ 2)

· 2 (bj−aj) qj+2

(qj+ 1) (qj+ 2)

= 4

n Y

k=1

(bk−ak)

2 (bi−ai) pi₍_b

j−aj) qj

(pi+ 1) (pi+ 2) (qj+ 1) (qj+ 2)

where we have utilised (2.7). Further, by (2.6), we have:

|Dn(f, g)|

≤ 1

2

" n X

i=1

LiHi−2 n Y

k=1

(bk−ak)

2 (bi−ai) pi+qi

(pi+qi+ 1) (pi+qi+ 2)

+ 4

n X

i6=j i,j=1

LiHj n Y

k=1

(bk−ak)2

(bi−ai) pi₍_b

j−aj) qj

(pi+ 1) (pi+ 2) (qj+ 1) (qj+ 2) 

 

=

n X

i=1

LiHi

(bi−ai)pi+qi

(pi+qi+ 1) (pi+qi+ 2)

+2

n X

i6=j i,j=1

LiHj

(bi−ai)pi(bj−aj)qj

(pi+ 1) (pi+ 2) (qj+ 1) (qj+ 2)

and the result (2.6) is thus verified. The sharpness follows from the sharpness of the Chebychev functional forn= 1 (see for example [12]).

Corollary 1. Letf, g:

¯

a,¯b

→Rbe Lipschitzian with constantsLi,Hi >0. That

is,

|f(¯x)−f(¯y)| ≤ n X

i=1

Li|xi−yi|, x,¯ y¯∈

¯

a,¯b

and

|g(¯x)−g(¯y)| ≤ n X

i=1

Hi|xi−yi|, x,¯ y¯∈

¯

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Then the inequality

(2.8) |Dn(f, g)| ≤

1 12

n X

i=1

LiHi(bi−ai)

2 + 1

18

n X

i6=j i,j=1

LiHi(bi−ai) (bj−aj)

holds and is sharp.

Proof. Takingpi=qi= 1 fori= 1,2, . . . , nin (2.6) readily produces (2.8).

Remark 1. Result (2.8) is presented in[12, p. 305], however, the coefficients of the sums are interchanged. Further, it is apparent that forx, y∈[a, b]andf absolutely continuous, then

f(x)−f(y)

x−y

≤ sup

z∈[a,b]

|f0(z)|=L,

demonstrating that a function satisfying a Lipschitzian condition is a weaker con-dition than one whose derivative belongs to L∞.

Remark 2. If n= 1, then forf, g∈[a1, b1]→R (2.9) |D1(f, g)|=|T(f, g, a, b)| ≤L1H1

(b1−a1)

p1+q1+1

(p1+q1+ 1) (p1+q1+ 2)

with

|f(x1)−f(y1)| ≤L1|x1−y1|

p1_, _|_g₍_x

1)−g(y1)| ≤H1|x1−y1|

q1

x1, y1∈[a1, b1]andp1, q1∈(0,1].

If n= 2then forf, g∈[a1, b1]×[a2, b2]→R

|D2(f, g)| (2.10)

=

1

(b1−a1) (b2−a2)

Z b1

a1

Z b2

a2

f(x1, x2)g(x1, x2)dx1dx2

− 1

(b1−a1) 2

(b2−a2) 2

Z b1

a1

Z b2

a2

f(x1, x2)dx1dx2

× Z b1

a1

Z b2

a2

g(x1, x2)dx1dx2

≤ L1H1

(b1−a1)

p1+q1

(p1+q1+ 1) (p1+q1+ 2)

+L2H2

(b2−a2)

p2+q2

(p2+q2+ 1) (p2+q2+ 2)

+2L1H2

(b1−a1)

p1₍_b 2−a2)

q2

(p1+ 1) (p1+ 2) (q2+ 1) (q2+ 2)

+2L2H1

(b2−a2)p2(b1−a1)q1 (p2+ 1) (p2+ 2) (q1+ 1) (q1+ 2)

,

with

|f(xi)−f(yi)| ≤Li|xi−yi| pi_{, x}

i, yi∈[ai, bi], i= 1,2,

and

|g(xi)−g(yi)| ≤Hi|xi−yi|qi, xi, yi∈[ai, bi], i= 1,2.

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(b−a)2

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[12] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´C and A.M. FINK, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.

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School of Communications and Informatics, Victoria University of Technology, PO Box 14428, Melbourne City MC, Victoria 8001, Australia

E-mail address:[email protected]

URL:http://sci.vu.edu.au/staff/peterc.html

URL:http://rgmia.vu.edu.au/SSDragomirWeb.html

Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, 41000 Zagreb, Croatia.