NATIONAL BUREAU OF STANDARDS REPORT

2977

On an Inequality of Hardy, Littlewood and Polya

by

A. J. Hoffman

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NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU QW SJAMBAUBS REPORT

MBS PR@JE€T MBS REPORT

1102-10-5116 December 17, 1953 2977

On an Inequality of- Hardy , Little wood and Polya

by

A. J. Hoffma:

This work was supported (in part) by the Office of

Scientific Re search f USAF

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BIREAM OF STANDARDS

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unless permlsi 25.D.C. Su<

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h a report has been speclfiilar report for Its own use,

On Inequality of Hardy , Littlewood and Poly

by

A. J. Hoffman*

A well-known inequality of Hardy, Littlewood arid Polya

(see L 3 3 ? P<, 4-9) st at ess Let o< = (^ ,*

.

9 ,c< ^ ) and ft =

xLE

• * • ? ft

.

2

be points in: R

n

, and let <r and T be permutetions of the set 1,2,... ,

n

such that > <?\ > > = 2 - - (TTi

and ^12 4

° 8 • ' (/:> f n

* Then in order that

/3 , < sr , ri * <5 i

ft> . + 'ft z \

> < c* r2 = <ri

+ V2

(1) + ^T2+ ...+ oUn_ u 1 + V2 + . . .+ W_ \ u \ n— i

;

a + ft + „ + ft = °<

, + « + + . T 1 c2 r n 0*1 cr 2

* o~n ?

it is necessary and sufficient that ft = X <X

, where X = (x. ..)

is a matrix satisfying

(2) x . . >0 for all i , j

,

(3) f

x

ij = I

x

ij

= 1 fo? a11 i’j*

Several proofs of this theorem, and of various generalizations, are known (see [l], [2], [4] and [5] for some recent

work). The purpose of this note is to present a strengthened

*This work was supported (in part) by the Office of Scientific. Research, USAF.
- 2-

version of the Hardy, Littlewood and Polya theorem (which

we call HLP hereinafter), the proof of which is no more

difficult than the proof of HLP.

We shall prove:

Theorem 1. Let o( = , o<2

, . . .

, c<^) satisfy

(M) <*, > w o >...>« .

Then in order that a vector ft = {ft j , . .

.

,/3 ) satisfy

( 5 ) X ft 2

’ •*

and

( 6 )

+ ^2 + -"+ ^„-1 +<t

2 + • • • + or n1

^-1 + ^2 + + ^n x

*1 + «2 +

it is necessary and sufficient that

(7) P = X<* ,

where X - (x. .) is a matrix satisfying (2), (3) A J

(3) x

i.

( j = x

i

2

j

fo2> a11 » i

2

? j such that ( j-i-j )

( -i

2

) >. 0,

and
-3-

( 9 ) x. . = x . . for all i, j

.

ij ji.

Proof of necessity* Consider the set of all vectors

/3 satisfying (5) and (6) . They clearly form a bounded,

closed, convex set in ft

81

* Let us designate this set by K,

iVe first show

Lemma . Let 1 1 a, < = n. Let y, = y^

= . • o=y F

= (tf-l + +eos+ %

^fli+1 + %1

+2+,eo

)/

a

1 ,y a.j + 1 y a, + 2

9,0 y a 2

)/ (

a

2-a

1 ) , . . .

,

y&

k_^ + 1 = ¥ a k

_

^ f

2

y a

k " ^ Qr ak _ 1 + I + ^Vl+ 2

+# ** + %)/(ak

a k-1 } *

Then (i) y

= (y-j ?y 2

? • • • »y ) is a vertex of K, and (ii) every

vertex of & arises in this way.

Proof of lemmas (i) First, we must show that y 6 K,

It is immediate that y satisfies (5)« To show that y satisfies (6), we first remark that the last equation of (6)

obviously holds, so all that needs to be established are

the inequalities. Further, if we prove

(10) y. + y 2 + °«,+ y s G«

, ? ?

for 1 1 s 1 a.j , it is clear from the definition of y that

this argument will establish (10) for any value of s, Now
(10) is equivalent with

(11) s(^j + a

2

+. . .+ a£ )/a

1 < ^ + ...+ <2 f g

.

But by (4), +...+ 1 ^

+ a

2+...+ %

+ (a l' s)

Hence,

( 12 ) s (tf.j+ . . .+ <* a

)/a-j 4^1 +# * #+c

^<3 + ( a

i _s )°*

s

- “— (°\ + ...+ * + (

a

1 -s )<* ) a

1

l s is

< <X, + +

< cs, , ,<v s 1 • • • ^ >

0f

s

( a -j - s ) - ( s + a

i - s ) ]

(by b)

which proves (11). v

To show that y is a vertex, assume that y = (y* + y")/2

where y ' , y" € K. Then if y' +y' > y 1 +...+ y„

=

I a -j

1 O -j cx

1 +...+ ck

, we would have y., n+...+y o " < *

I Ct-| & O -j1

'***

-j } violating (6). Similarly y^’ +...+ y'

fl — y"..

+...+ v

M y a. = y 1 + -*-+ *V’ J =

J j

Next, suppose for some s , 1 < s < a, , we have y' > y o

.

I s s Then by the preceding paragraph, there is an integer t,

1 < t < a 1 such that y' < y . Note that y = y . Hence

y* > y\. If t < s , then y' violates (5). If t > s

,

y" violates (5). Hence
y« = y" = v _ e Clearly this argument extends to any

" S " s §

s = Thus y = (y ' + y

n )/2 implies y = y' = y".

This completes the proof of (i).

To prove (ii), let /3 satisfy (5) and (6), but not be a vector such as y above. This means that there exists integers I < <...< fl|, = n such that /3- = = «»* = h, < V+ 1 = * ”= <•••< ^a. 4"*=/5 ” ’ bHt for a? x— s ® 2 ‘k one j , ... j b i , we have ^^

i 2

^ .

w + . # .+ _ .Let £ be any positive number smaller than

c. a •

(b „ < , +

a . ']

o( + . 0 . + ~ & _ « ~ $

-4 O • 9 a

. ' 7 ' a a

? ^4 i a . a

J J

" o-c'-ie 0 '=n ,...,/g' = t . /»’ — /O f

; +

_ /3 + £ /(‘a .-a - . ) «, /S*

3-1

a +1

j

“ 7 -1 «. -tl /3

{

' J a ,

. ~

o * , a. ca o , - i-i- 1 i+

i/(a . _.-a,) , '5’ . j + i a.M+l

? *T 1

/? r

• • 5 6

' = 13 n

/3” = 2 fi> - fi

x . Then it is easy to check that and %" each

satisfy (1) and (6) , /5 = (/3 • + /3")/2, /3

1 ¥ ft"

,

so /? is not

a vertex of K. This completes the proof of (ii), thus of

the lemma.

We resume* our proof of the necessity of theorem 1 . Let

(b £ K, then ft is a convex combination of the vertices of

K. By the lemma, each vertex of K is of the form x . , . »

,

7 a

'

, where X , . . .

,

is the matrix of order hi

' k

a

1 a

k
- 6

-

with O's everywhere outside the square boxes along the main

diagonal, where the first box is of order a^ , the second

box of order a^-a^,..., the k-th box of order a

jc _i 5 every entry in the first box is 1/a^, every entry in the

second box is l/Ca^-a^ )

, . . » , every entry in the k-th box

is 1/(a ( ,-ar

j ) . Note that Xa ,..., a satisfies (2), (3)>

1 k ci ^o) (3), (9). Since /3= 2^c.(X (l) o< ), wher<y?c . = 1, X

(i) is

a matrix of the type (12), we may write (3 = ( ^

c^ X^^)cx,

and observe that any convex combination of matrices satisfying

(2), (3) ? (3), (9) also satisfies these conditions. If

v1 (i) we set X = Zj c

± X , the proof of the necessity is complete.

Proof of sufficiency ; We shall show that if a matrix

X satisfies (2), (3), and (3)-— (9) is not needed — , and a vector o( satisfies C^), then - X<* fulfills (5) and (6).

We first show

(13) P

±

> 0i+1 (i = 1,...,n-1)
-7-

For \ - /3 i+1 = 2 Zj Ofj, where Zj = - x i+1>j . By

virtue of (8), z. > 0 for j < i and z. < 0 for j > i + 1.

J ~ ~ J - “ By (3) , 22 7. • = 0. It follows from (4) that if we set

J u = z-j +...+Zj and v = Zj +1 +,..+z

n = -u, Z z . o<

j

j>

+ v ^i+1

= ^ 0a This proves (5). The

proof of (6) - which depends on (2) and (3) ~ is the same

as in HLP.

Bibliography

[ 1 ] R. Bellman and A. Hoffman, "On a Theorem of Ostrowski

and Taussky," to appear in Archiv der

Mathematik.

[2] D 0 Blackwell, "Equivalent Methods of Comparison," Ann.

Math. Stat., June 1953.

[3] G. Hardy, J. Littlewood, and G, Polya, "Inequalities,"

Cambridge, 1934.

[4] A. Ostrowski, "Sur les Fo net ions Convexes et Concaves au

sens de iseh'Air,. Jour, de Math, Pares

et Appl . , 31(1952), p. 253.

[5] S. Sherman, "On a Theorem of Hardy, Littlewood, Polya,

and Blackwell," Proc. Nat. Acad. Sci.,

37(1951), p. 826 .

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