Chapter 28 The Multiple Cauchy Integral and the Cauchy Inequalities

CHAPTER 28

THE MULTIPLE CAUCHY INTEGRAL AND THE CAUCHY INEQUALITIES

(The Multiple Cauchy Integral).

PROPOSITION 28.1 separated,

f E #(U;F),

nl,

N,

...,nk E

that j

m = nl

+...+

+...+hkxk

5 + hLxl

= l,,..,k.

5 E

k

nk

and

E U

xl,.

E,

> 0, such

P 1 , ...,pk

for every

..,xk E

be

lhjl

E 6,

P j

Then n

..%!

1

nl!.

-

N",

E

U,

F

Let

dmf(g)xl

f(5+hlX1+.

1

(2ni)

k

...xkn .

l

x1

-

-

.

.+hkXk)

nl+l

*/hjl"Pj

k

nk+ 1

dh l...dXk

"'h,

1sj s k REMARK 28.1: The proof of Proposition 28.1 is similar to the

proof of Proposition 25.2, the single integral in the latter case being replaced by a multiple integral.

Alternatively,

Proposition 28.1 can be obtained by repeated application of Proposition 25.2.

Proposition 25.2

Proposition 28.1 in which where

nl =...=

k = m,

COROLLARY 28.1 m E N*,

x1

,...,xm E

g + xlxl +...+ j = l,...,m.

Let

xmxmE Then

k

is the extreme case of

= 1. The other extreme case,

nm = 1

is as follows:

F

be separated,

E

and

u

p1

,...,pm

for every

xj

5

f E a(U;F),

E U,

> 0 such that E C,

IxjI

4

p j

9

9

2 30

CHAPTER 2 8

Xx = XIXl and if

+...+

IX,I

...,nk) E

n = (nl,

If we write

REMARK 2 8 . 2 :

'kXk

= pl,

Xn+l

'

...,I

hkl

nl+l

= X,

= pk

...Iknk+l ,

i s written

k

N

,

d ( n ) = k,

dX = dX,.

1x1

= p ,

..dx,

,

then

integral formula given in Proposition 2 8 . 1 becomes f

a form similar to Proposition 2 5 . 2 . REMARK 2 8 . 3 :

sition 2 8 . 1 ,

Cauchy inequalities can be derived from Propoo r from Remark 2 8 . 2 ,

in the same way as the

Cauchy inequalities of Chapter 2 5 were derived from Proposition 2 2 . 2 . REMARK 2 8 . 4 : A E Ss(%;F),

If we apply Corollary 28.1 to

5 =

with

larization formula:

A(xl,

...,xm) =

and

0

f1

1

m!(2ni)m

U = E,

f =

i

(XI*

where

we obtain a new po-

m xm )

2(X1Xl+...+h

.$ jl=l

i,

-__ 2

*Xm)

dX1. .dX,

1s j 4 m

where we have taken

p1

=...=

P,

= 1;

in this case, it can

be shown easily by a change o f variable that any choice of

t

THE MULTIPLE CAUCHY INTEGRAL AND THE CAUCHY INEQUALITIES

> 0 gives the same value for the integral.

p l , ...,pm

231

Like

the original polarization formula (Lemma 15.1), we can use this formula to obtain an estimate f o r a(xj)

r;

1

REMARK 28.5:

to

f =

i,

j = l,...,m,

for

BIA(xl,

...,xm)];

then

More generally, if we apply Proposition 28.1 where

A E Xs(%;F),

m E N",

5

= 0

U = E,

and

we obtain another new polarization formula:

n

k k -

. -

if

1

k

m! ( m i )

!Ixjl=l 1sjsk

n +1 1

1,

"'1,

nk+ 1

dX l...dXk

.