GH HARDY and JE LITTLEWOOD, Cambridge, England. 1

Some More Integral Inequalities,

by G. H. HARDYand J. E. LITTLEWOOD,Cambridge, England. 1. This is in a sense a companion to an earlier note with a similar title(1) ; and the contents of both notes were suggested by a more systematic investigation concerned primarily with series. Both in the note just referred to and here we are concerned with special cases of a general theorem which we have stated without proof elsewhere(2), and also (and more particularly) with their integral analogues. These special cases are peculiar, and deserve special treatment, because in them we can go further than in the general case and determine the best value of the constant K of the inequality. The actual inequality proved here, which corresponds to the case p = q = r of " Theorem 25 ", is much easier than that proved in Note X(3), but has some distinguishing features. There is generally an attained maximum, which is unusual in such problems ; and the proof that the maximal solution is effectively unique involves points of some independent interest. Integral theorems. 2. In what follows it is assumed throughout that f and g are positive (in the wide sense) and measurable. Theorem 1. If and (2.1)

then (2.2)

where

(1) G. H. Hardy and J. E. Littlewood : "Notes on the theory of series (X) ; Some more inequalities", Journal of London Math. Soc. 3 (1928), 294-299. (2) G. H. Hardy and J. E. Littlewood : " Elementary theorems concern- ing power series with positive coefficients and moment constants of positive func- tions ", Journal fur Math., 157 (1927), 141-158. The theorem in question is Theorem 25; the proof is very intricate, but we have now written it out at length and hope to publish it shortly. (3) see (1). 152 G. H. HARDY AND .J. E. LITTLEWOOD

(2.3)

Theorem 2. There is equality in (2.2) if and only if (2.4) where A, B, c are positive. An alternative statement of Theorem 1 is Theorem 3. then

(2.5) with

(2.6) and the same K.

It is to be understood that when, as here, we omit the limits of integration, they are 0 and •‡. We require this form of the theorem for comparison with the case p=1. If p=1, Theorem 3 becomes

Theorem 4. If l•…1, m•…l, and not both 1 and in. are 1, then

(2.7) where

(2.8) unless f or g is nul. The constant is the best possible. It is to be understood that, e.g., (1-1)1-l means 1 when 1=1. Theorem 5. If 1=m=1, there is equality, with K=1, for all f, g. Theorem 4 is the limiting form of Theorem 3 ; it is easily veri- fied that (2.8) is the limit of .(2.3). A formal passage to the limit in Theorem 1 yields only the entirely trivial Theorem 5. The only one of the theorems whose proof is not quite easy is Theorem 2.

Proofs of Theorems 1, 3, 4, 5, 3. If

so that SOME MORE INTEGRAL INEQUALITIES. 153

where then

(3.1)

(3.2)

This proves Theorem 1, and Theorem 3 is equivalent. To prove Theorem 4 we write

when

and so

where

There is inequality unless

for almost all x and then for almost all u. This means that

and so that one of ƒÓ and ƒµ is nul.

To prove K the best possible constant, we take ƒÓ=1 when

and ƒÓ=0 otherwise, and define ƒµ similarly (with ‚• for 1). Then

where es tends to 0 with ƒÂ, uniformly in x. The result follows when

we integrate and suppose that ƒÂ•¨0.

We have thus proved Theorems 1, 3 and 4 ; and Theorem 5 is. trivial. 154 G.H. HARDY AND J. E. LITTLEWOOD :

Proof of Theorem 2. 4. There will be equality in (3.2) if, and only if, there is equality in (3.1) for almost all x ; and there will be equality in (3.1), for a given x, if and only if is constant for almost all u. We describe a sot whose complement is nul, for shortness, as a full set, and we have then the following necessary and sufficient condition for equality : (A) there is a function X(x), defined in a full set X; and to every x of X corresponds a full set U(x) in which (4.1) We have to deduce (2.4), which is equivalent to (1) (4.2) from (4.1). In the argument which follows capital letters X, U, T...., with or without subscripts, denote full sets depending on any parameters which may be indicated ; and e denotes the relation of inclusion in a set. Thus ue U(x) means that u belongs to a certain full set U which varies with x. . The arrow •¨ is Hi1bert's symbol of impli- cation. Thus (4.1) will be written

5. We show first that (A) implies: (B) there is a function Q(x),defined for all x, such that

(the last equation is true for every x and a full set of u depending upon x). We have formally (5.1)

if These equations hold if

(1) With different A, B, c. SOME MORE INTEGRAL INEQUALITIES. 155 and

(ƒ¿) (ƒÀ) The common part of any number of full sets is a full set. Hence, for any x, there is a U1(x) such that

and to any u of Ui(x) corresponds a T1(x, u) such that

In these circumstances (ƒ¿) is true, and the sets referred to in (ƒÀ)

all exist.

Next, taking u in U1 and t in T1, the first two of the relations

(ƒÀ) are satisfied if

the common part of U(u) and U(x-u + t) ; and the last two if

Finally V1 and V2+ t (the set obtained by translating V2 a distance t) have a common part. Hence, when u, is in U1(x) and tin T1(x, u), it is possible to choose v so that all the equations (5.1) are true. Fix x, and u in U1(x); and let s=u-t. When t varies in T1(x, u), s varies in a full set S1(x,u); and, for s in S1(x,u),

exists and is independent of s. This proves (B). 6. Next, (B) implies that (C) for all x, u. In fact, if we denote by V(z) the full set of v in which, after (B),

we have

provided that v lies in V(v.) and V(O), -v in V(x-u), and u-v in V(x) ; and this is so for almost all v. 156 G. H. HARDY AND J. E. LITTLEWOOD :

7. Since Q(x) is measurable, it follows from (C), by a theorem of Siorpinski(1), that

where C and c are constant. Hence (7.1) for all x and almost all u. If we write

(7.2) then (7.1) gives (7.3) for all x and almost all u.

We define ƒµN and wN by

Then (7.3) gives

for almost all u ; and so, since ƒµN is measurable and bounded,

Since wN, is continuous, this is true for all x, t ; and t=x gives

Hence ƒµN, and therefore ƒµ, is null, and

It now follows that (7.4) i.e. that

for almost all x and then, for such x, for almost all u. If we write

(7.5)

(1) W. Sierpinski : "Sur 1'equation fonctionelle f(x+y)=f(x)+f(y)",Funda- menta Math., 1 (1920), 116-122. (2) With a different C. SOME MORE INTEGRAL INEQUALITIES, 157

then (7.4) becomes (7.6) We define ƒÏN and ƒÐN as we defined ƒµN above, and write

Then, from (7.6), and so (7.7)

This is true for all x, t ; and t = x gives

and so (7.8)

Since RN(x) is continuous and satisfies (7.8), it is a multiple of x,

say CN x ; and ƒÏN•ßCN, which is plainly possible only if CN is inde-

pendent of N. Hence ƒÏ and ƒÐ are (almost always) constant, and

(4.2) follows (7.5). This completes the proof of Theorem 2.

Series.

8. Suppose (as in Note X) that an•†O, bn•†0, ao=bo= 0, and

for Then the theorem for series corresponding to Theorem 1 is Theorem 6. If p>1, 0

where K is defined , by (2.3), unless an=0 for all n or bn=0 for all n. If a>1 or b>1, the result is still true for some K=K(p, a, b). In fact, if

we have

If 0

(8.1)

Hence

unless for all n, i. e. unless an = 0 for all ii or ƒÀn=O for all n. This proves

the first part of Theorem 6(1).

If a > 1 or b > 1, (8.1) is no longer true, but

for some K, and from this we deduce the second part of the theorem. There is no analogue of Theorem 2, and those of Theorems 3, 4, and 5 may be left to the reader.

Generalisations. 9. Theorem 1 generalises to 3 Theorem 7. If

then

The proof is an obvious generalisation of that of Theorem 1. There are naturally corresponding generalisations of Theorems 2-6.

Fourier Faltungs.

10. A less obvious extension is that to the range (-•‡, •‡), with

Theorem 8. If

(I) This corresponds to Theorem 1 of Note X. The conditions 0

with

The proof is similar to that of Theorem 1, the part of the integral

being now played by the integral

The additional condition a + b < 1 is required to secure the conver - gence.of this integral. There is no analogue of Theorem 2, and we have not proved that K is the best possible constant. There is a corresponding analogue of Theorem 3 of the pre- vious Note X, in which the interval is (-•‡, •‡), h is defined as in Theorem 8 above, F =f/p and G= g/q satisfy the additional condition F+G<1., and