I ntrnat. J. Math. Mh. Sci.
Vol. 5 No I(1982) 141-157
141
SPECTRAL INEQUALITIES INVOLVING THE SUMS
AND PRODUCTS OF FUNCTIONS
KONG-MING CHONG
Department of Mathematics, University of Malaya
Kuala Lumpur 22-11 MALAYSIA
(Received April 2, 1975 and in revised form December II, 1980)
In this paper, the notation
ABSTRACT.
<
denote the Hardy-Littlewood-Polya
and
spectral order relations for measurable functions defined
(X,A,) with (X)
spectral inequalities.
log[b +
L
1([0,
ment.
(flg) +]
a]),
here
and expressions of the form
a,
(fg)+]
log[b +
and
LI(x,A,),
f, g
If
I
With the particular
log[b
+
s
t
on a fnite measure space
f- g and
b
fg
Lburg spectral inequality fg
are called
proven at, for se
(fg)+]
enever log+[b
respectively denote decreasing and
case
f ’, g
+
(where
g
lity.
f, g
LI(x,A,)), thus
(fg)+]
ncreas
rearrange-
0 of this result, the Hardy-Littlewood-Polya-
for
0
f, g
LI(x,A,)
is
s
consequence of the well-o but seingly uelated spectral inequality
+
giving new proof for
e
foyer
(flg) +
not necessarily non-negative
(fg)
f, g
E
f
spectral inequa-
+
+
to be a
f + g
reover, t rLittleod-Polya-L5urg spectral inequality is
tended to give
O,
b
also
(fg) and (fg)- (fg)- (flg)LI,A,B).
for
,
Y R D PS. Esle Hets, Spl Inelies, nve
dete m, nc m, gle cpe or.
1980 TICS SCT CLSATION CODE.
I.
26D20.
INTRODUCTION
In this paper, we prove some spectral inequalities involving the sums
an prod
ucts of measurable functions as well as their equimeasuraSle rearrangements,
W,tk
K.M. CHONG
142
these results, we extend some knownrearrangement inequalities of Day
[2,
Hardy-Littlewood-Plya
Theorem 378, p.
[4, Proposition i,
ntz-Shimogaki
34]
p.
278],
[3,
London
[5,
and Luxemburg
[i, pp.941-943],
Theorems i and
Theorem 8.2, p.
2], Lore102].
Moreover, using our method of approach, we obtain in several cases more general conditions for equality than those given by either Day in
2.
or London in
[3].
(X)
a.
PRELIMINARIES.
(X, A, )
Let
X
[I
be a finite measure space with total measure
.d
is clear from the context, we often write
function with either the positive or negative part integrable.
the set of all extended real valued measurable functions on
M(X,)
and
(written
f
g E M(X’
), where
’(X’)
for all real
If
If
t.
R
/
a, are said to be equimeasurable
(X)
g, it is not hard to see that
f
f
(2.2)
(g)
f
on the interval
and
f(s)
s e
[0,a].
decreasing function
If
f
called the decreasing reagement
[0, a],
In fact,
are equimeasurable.
inf {t e R:
f(x) > t})6 s}
({x
(2.3)
Observe that there also exists a unique right continuous non-
-6_f,
If
called the increasing rearrangement of
f,
such that
R
by
m.
f.
In what follows, we denote the Lebesgue measure on the real line
It is easy to see that
1,2,3,
t" f
f n n=l
whenever
f
+
nn=l
where
f,
f
n
f
M(X,I.I), n
The following theorem generalizes this fact.
THEOREM 2.1
If
f
n
e M(X,),
where
61ira
and
f E
f e M(X,B), it is well-known that there exists a unique right continuous non-
f, such that
for all
Two functions
is a Borel measurable function.
R
increasing function
of
we denote
’({x:g(x) > t})
f(x) > t})
(f)
:
X.
M(X,)
By
for any
g) whenever
B({x
whenever
X
for integration over
Whenever
n
inf f n
(t)
1,2,3,
lim inf
then
6f n (t)
(2.4)
SPECTRAL INEQUALITIES OF FUNCTIONS
Slim
for all
t
e
[0,a].
6f
as
Thus, if
sup f
f
(t-)
6fn
(t-)
-a.e.
as
lira sup
n
converges to
n
f
143
(2.5)
n
/
f n con-
then
0%
6f.
then
Furthermore, if fn converges to f in measure
fn also converges to 6f
P
in measure and
fn converges to f in L if and only if f converges to 6f
in LP, where 1
p <
verges to
n
at every point of continuity of
/
.
PROOF.
ltuation (2.4)
f n n--I/ f
follows from the fact that
whenever
/ f
fn n--I
in exactly the same way that Fatou’s Lemma is obtained from Lebesgue’s Monotone Conve-
Next
rgence Theorem.
fact that
6_f(t)
((a- t)
,<
Nw’
{6fn
ence of
sequence
t E
[0,a],
n
}’n--1
}; then
I1
fni j
fni
f
[0,a], when=e (2.5)
in measure as
n
/
is a subsequence of
i=l
which converges to
f
’
in measure (cf.
li---
6f n
follows.
let
{fn }
fni
and
be any
subsequ-.
i=l
hence contains a sub-
-a.e., by Riesz’s Theorem. Then
j=l
fni
also converges to
and
j--i
f m-a.
j--I
so by Riesz’s Theorem again, we conclude that
/
and using the
we immediately obtain
InI
cnverges t
j--I
j
for
t) ), t
is a subsequence of
fnij
e. as
f
t)-)
-6f((a
l-m f n ((a-
if
{-f
on applying (2 4) to the functions
6fn
converges to
f
[6, (11.45), p.163]).
The last assertion follows directly from the preceding paragraph and the fact that
the operation of decreasing rearrangements preserves uniform integrability (see
Section
2]).
REMARK.
The preservation of
L
convergence is also shown to be true in
[7,
K.M. CHONG
144
COROLLARY 3.5 below using a different method of proof.
M(X’
M(X,)
If
f, g
’(X’)
a <
,
then we write
0
and
f
"
f
and
f.<’
+
g
E
I
L (X,)
I
L (X’
,
(X)
where
whenever
g
fdm
+
dm
[O,a]
t
g
0
(2.6)
f- g and
whenever
g
,
By an argument given in [8, p.152], we infer that (2.6) is equivalent to requiring
t)
f
t)
+
(2.7)
Moreover, it can be shown (see [9, Proposition 10.2 (iii), p.62],
t e R.
for all
I(g
+
for example) that
II f ILo II II=o
f
whenever
g
L (X,)’L (X’,’).
g
where
(2.8)
In the sequel, expressions of the form
g (respectively
f
f.<-< g)
are called
trong (respectively weak) spectral inequalities
3.
SPECTRAL INEQUALITIES INVOLVING THE SUMS OF FUNCTIONS.
In this section, we give simpler proofs for the spectral inequalities obtained by
Lorentz-Shimogaki [4, Proposition I, p. 34]
[i, p. 941] and we also extend
and Day
them to include not necessarily integrable functions defined on a finite measure space
In what follows, if
x
(x I, x 2,
x
n)
x
and
n)
(Xl,
x2,
x’
(x I, x
x
easing and increasing rearrangement of
x’)
2,
x
e Rn is any n-tuple, then we write
where we have regarded
function defined on a discrete measure space with
LEMMA 3.1.
a
If
(al, a2,
,
n
E (a i +
i--I
PROOF.
If
I
b)
+
n
a
n
n
Y.
to denote respectively the decr-
n
(ai+ bi
i=l
R
+
n
and
n
Y. (a
i=l
n
and
bi <
bj,
as a measurable
atoms of equal measures.
’= (bl, b2,
, ,+
+
i
b
n)
R
n,
then
bi)
Without loss of generality, we may assume that
i < j
x.
a
i --ai,
then it is not hard to see that
i
I, 2,
(ai+ b i)
+
n.
+
(a.+3
SPECTRAL INEQUALITIES OF FUNCTIONS
bj)
+
bj)++ (aj+ b i)+.
(ai+
<
145
i, j
Thus, for each pair of integers
I
such that
i
the middle sum of (3.1) is never decreased on interchan-
< j
n,
whenever
b
ging
b.
and
We therefore conclude that the left hand sum is the minimum possi-
b..
i
bj,
value while the right hand sum is the maximum possible value attainable by the
runs through all its rearrangements.
middle sum as
LEMMA 3.2.
I
f, g e L (X,) then
If
Ig)+dm
(f+
0
+
(f + g)
(6f
0
d
X
+
+
g
(3.2)
dm.
We first note that it suffices to prove (3.2) for non-atomic measure spa-
PROOF.
[9, pp. 52-54]
ces by imbedding (X,A,p) in a non-atomic measure space; we refer to
or [5] for this method.
(X,A,)
Thus, we assume that
is non-atomic.
To prove (3.2), we need only prove the right hand inequality, the rest is similar.
To this end, we first show that there exist sequences of simple functions
and
{gn
i} such that both
n
rywhere but also in L
preserving map
For each
n
o:X
/
I.
fn
/
f
(X,A,p)
Now since
[0,a] suck that f
I, 2, 3,
o-l([
n
g
i, 2, 3,
(i-l)a
i__a ])
n
n
Theorem
2
n
6.2, p. 49].
let
2
I}
n
by
2n
n
For each
n
i, 2, 3,
2 n}
i -I, 2, 3,
)(A)
IA
f dll
A
i=l
(Ei (n))
let
A
and let
A,
A
E.
i
denote the
o-algebra generated by
then, by the martingale convergence theorem
56, p. 369],
f
d(,IAn
n
d(ll JAn)
{E (n)
o- algebra generated by
denote the
f
I}
not only pointwise almost eve-
-a.e. [9,
o o
i
2
Define a sequence of functions {f
/
n
n
is non-atomic, there exists a measure
f
and for each
E.(n)
x
_gn
and
{f
-a.e.
[6,
nffii
An"
If
Theorem 20.
146
as
K.M. CHONG
,
n
where the convergence takes place point-wise almost everywhere with the
()
where
9, p. 70],
oo is
B denotes the Borel o-algebra of
[0,a]
f
n
in
L
i}
f
1
f
n
as
n
/
.
gn
n
since
n
I}
/
0
of the fact that
fn
n
/
Theorem 5.1, p.
403],
-I
f
n
{gn
/
Ifnl i<
Ignl
Ign’l
n
/
f’n
But then
oo.
f’n fn
and both
/
f
and
-
gn
o% and the convergence also takes place in
f’nl -< fnl
403] since
f --:>
o
there exist two sequences of simple functions
pointwise almost everywhere as
Theorem 5.1. p.
[7,
Aoo
[9, Proposition I0.
But, by
and so, by
If’In Ifnl’
pointwise almost everywhere as
[7,
measurable and
having the same sets of constancy and with these sets
having equal measure such that
gn gn
Aoo-
Similarly, there exists a sequence of simple functions
I
+ g pointwise almost everywhere and also in L
as n
{g
and
f
I, 2, 3,
[I, Lemma 5.5, p. 938]
i}
n
f
for
such that
Now, by
{f’n
-a.e.
f
limit function being
J
fl
and
g =>
Ifl
and
gn.
gn’)+
d
I (f,
g
nl
-<
Ignl
g
nl <
Igl
gl
and
+
L
g
I
by
by virtue
[9,
(see
0
Theorem
I0.I0, p. 71]).
Hence, by Lemma 3.1, we have
IX
(f’ +
n
a
0
+
g
n
)+
dm
where the result follows by taking limits.
THEOREM 3.3.
If
LI(x,),
f, g
6f
PROOF.
On substituting
+
Ig
/
f- t
f
then
+g< f
for
f
+6
(3.3)
g
in (3.2), we see that the result follows
immediately from Lemma 3.2, by virtue of (2.7).
REMARK.
and
[5,
The spectral inequality
Theorem i0.i, p.
108])
f + g
-< 6f
+ 6
and it can be used
is well-known (see
g
[7,
Theorem 2.2, p.
[9, p. 88
397]
for
giving a simple proof of a result concerning the uniform integrability of functions.
On the other hand, the spectral inequality
p. 941].
f
+
ig-
f + g
is obtained by Day in [i,
In either case, our present method of approach is simpler.
SPECTRAL INEQUALITIES OF FUNCTIONS
COROLLARY 3.4.
If
f, g
g
LI(x,),
6
6f
PROOF.
Since
stituting
-g
g
for
COROLLARY 3.5.
i], m)
-6_g,
Ig
f
g
147
then
g
(3.4)
f
the result follows immediately from Theorem 3.3 on sub-
(3.3).
in
[4, Proposition I, p. 34]).
(Lorentz-Shimogaki
LI([o,
f, g
If
are positive, then
This is an immediate consequence of the preceding corollary and a theorem
PROOF.
of Luxemburg
REMARK.
[5, Theorem 9.5,
The notation
f
-<
p.
107].
g
used by Lorentz and Shimogaki in
[4] means
in our present context.
Suppose
COROLLARY 3.6.
to a function
f
L (X,)
f
as
n
g
L (X,), n
I, 2,
n
/
oo, then
f n also
If
f
n
converges in
converges in
L
f
to
L
as n
/
oo.
Using Corollary 3.4, we have, for each
PROOF.
f
"<.- II f n
..llf f[loo
by (2 8)
and so
f[Ioo
n
f n f fn-
I, 2, 3,
whence the result follows.
The following simple theorem is useful especially for generalizing certain rearrangement inequalities (by removing restrictions that certain functions must be non-
negative) and certain spectral inequalities (by removing restrictions that certain functions must be integrable).
THEOREM 3 7
n)+-n
satisfies
ffndn__+0 If
ntly
In general, if
dta
+
f dla
If
f
]fn
LI(x,)
if]
then the sequence
for all
{f
n
f
defined by
n=O
n
and decreases to
g
I
L (X,), then, as
f
as
n
oo.
n
(f +
Conseque-
d.
f
M(X,V)
with
f+
n + oo, f
n
+
f
and
f
n
R.
n 0
PROOF.
The result follows immediately from Lebesgue’s Dominated Convergence Theo-
K.M. CHONG
148
rem. and Levi’s Monotone Convergence Theorem.
The following theorem extends Theorem 3.3 to include measurable functions which
are not necessarily integrable.
THEOREM 3 8
Suppose
f, g
f
f+
or, equivalently, if
+
(f + n)
lg n
(lg
lg n
f
n
n,
+ n)
+
gn
Theorem 3.7.
gn
+
(g + n)
g
f
n,
-
f
+
But, as
Hence
f + g,
+
Ig
Suppose
g
I
+
e L (X,),
then
g
(3.6)
"f
Ig"
(3.7)
f + g,
Ig
where
by Theorem 3.3.
f
+
the rest is similar.
0, i, 2,
n
fn’ gn e LI(x’)
Clearly,
n.
COROLLARY 3.9.
n,
+
<f
g
I
If suffices to prove
+
f + g
Ig
f+
If
L (X,), then
g e
f
PROOF.
M(X,)
g
n + oo,
fn
+
f n (f
Then
for
O, 1, 2,
n
f
ig n
+
Let
Ig,
fn
+ n)
and so
fn + gn
n,
f
n+
f + g by
[I0 Corollary i.II].
by
f
fl’ f2’
g
M(X,)
f.+
If
L I (X,), i
g
i, 2,
then
fl + f2
PROOF.
+
fn f i + 6f 2
+
+
+
fn
(3.8)
The result follows from Theorem 3.8 by induction.
As a direct consequence of Theorem 3.8 above, the following theorem proves some
basic spectral inequalities which can be used to give simple proofs for some known
spectral inequalities involving the products of functions.
THEOREM 3.10.
LI(x,)
f, g
If
are either both non-negative or both non-posi-
tive, then
log
In general, if
If
f, g
f
fl’ f2’
log
PROOF.
f Ig<
g
LI(x,)
LI(x,)
g
n
flf2
log
log fg
f
n
-
log
(3.9)
fg.
are all non-negative, then
6f I f 2
(3.10)
fn
are both non-negative, then the result follows immedi-
ately from Theorem 3.8 on replacing
f, g
respectively by
log f, log g
and on using
149
SPECTRAL INEQUALITIES OF FUNCTIONS
log
the fact that
log
+
log+g
f +
log
g
log
I(x,).
LI(x,)
f, g
g
and
(log f + log g) + ,<
fg)+
(log
f + g g L
The case that both
ing the previous case to
4.
f,
log
f
SOME SPECTRAL
-g, and using
and
-f
are non-positive follows immediately by apply-
[9,
INEQUALITIES INVOLVING THE PRODUCTS
Theorem 4.6, p.
33].
OF FUNCTIONS
The following theorem extends some spectral inequalities obtained by Luxemburg
Theorem
10.4] (cf. [9, p. 88])
and Day
[5,
Theorem 8.2, p.
Theorem 378, p. 278]
of
to include not necessarily non-
Using this extension, we can easily derive Luxemburg’s
negative integrable functions.
generalization
[i, p. 942]
[5,
102] of
a theorem of Hardy-Littlewood-Polya
[2,
Furthermore, our method enables us to give a further extension
Hardy-Littlewood-Plya-Luxemburg’s
Theorem using increasing functions of mixed con-
vex-concave type (see Theorem 4.2 below).
Suppose
THEOREM 4 1
(flg)- Ll([0,a],
f
m)),
g
g
L
I(X,).
(f)+
g
g
L
1
([0 a]
m) (respectively
then
.
(flg)+’i
(respectively
If
(fg)
(fg)
(fg)+<-Q (fg) +
(flg))
(4.1)
and the strong spectral inequalities
(fg)+’ (fg) + (respectively (fg)- (fg)--i (flg)-) hold
+
(fg)- (flg)-)(respectively (f)(flg) (fg)+ (f)+
g
g
(flg)+
if
Furthermore,
[(fg)
[(flg)+] <’ [(fg)+] <
<’< [(flg)
+[(fg)+]
])
Ll([0,a],
[(f g )+]
for all increasing convex functions
m)
(respectively
+[(flg)
tion,
is strictly convex and increasing such that
pectively
[(flg)
[(fg)+]
and only if
PROOF.
(respectively
L
1
([0,a], m)),
e L
1
:
R
+
if and only
[(fg)-] <
R
([0,a], m)).
such that
If, in addi-
Ll([0,a],
[(fg)+]
then the strong spectral inequalities
m) (res-
[(flg)
-<.[(fg)+] (respectively [(fg)-] [(fg)-] [(flg)-]) hold
(flg)+ (fg)+ (fg)+ (respectively (fg)- (fg)- (flg)-)We first prove the theorem for the case that
case, (3.9) holds.
f
0
and
g
O.
+
if
In this
150
K.M. CHONG
Assume
LI(x,).
fg e
Since the exponential function
[I0, Theorem 3.3]
Again, if
log(l +
et),
R,
t e
fg ,’
The spectral inequality
then log(l + fg) e
LI(x, ll),
(3.9),
to the left spectral inequality of
6flg’< fg.
fg e LI(x,),
(log fg), i.e.,
fg e
exp(log fg)
ng and strictly) convex and since
LI(x,)
R
R
exp
+
is (increasi-
we have, on applying
6f6g
6flg)../’ exp
(log
exp
is proved analogously.
t+
and, since the function
is increasing and convex, by Theorem 3.10 and
[I0, Theorem 3.3],
6flg)"<’< log(l + fg). Thus, by [I0, Corollary 2.4], the strong spectral inequality 6flg-< fg or, equivalently, I + 6flg< i + fg, i.e., exp(log(l +
6flg)) exp(log(l + fg)), holds if and only if exp(log(l + 6flg)) exp(log(l + fg)),
1 + fg or, equivalently, 6flg
fg. Similarly, if 6fg e Ll([0,a
i.e., i + 6flg
m), then fg
6fg and fg 6fg are equivalent.
By [i0, Theorem 3.3], it is clear that (6flg) ’< (fg) (respectively (fg)
I
+
(6f6g)) for all non-decreasing convex functions : R R such that + (fg) L
(X,) (respectively +(f6g) e Ll([0,a], m)). The last assertion is an immediate
we have log(l +
/
[i0, Theorem 3.3] and the foregoing result.
consequence of
To prove the theorem in general, we need only establish the spectral inequality
(flg )+<-< (fg)+
and the corresponding assertions concerning the strong spectral in-
equality
(flg)+
(6fl g)+
f+
f+g+
+ f g
g
+
(fg)
+
f
+,
the rest is analogous.
6
(since
g
f+
and
If
f
and
.+
f
(fg)
+
f
where the summands on the right in each expression have disjoint supports
Now by what we have proved,
easily from (2.7) that
(6flg)+’<
6f
To this end, we first note that
(fg)
+
6f+
(6flg)
g
+
+ f+g+
f
+
6f+
6f+ +’< f+g+
f+
g
if and only if
+
f
g
f
g
+
f+g+
g
_’<
6
and
g
f+
(fg)
and that
But, by the re-
f g
f g
_<’ f-g-
6
and
f
g
+
g
f+g+
+
and
f
It then follows
f g
g
-< f+g+ + f g
suit established in the preceding paragraph,
respectively equivalent to
_<
6
and
g
are
151
SPECTRAL INEQUALITIES OF FUNCTIONS
8 +
Thus, using (2.7) again, we conclude that
g
f
ad
f+
only if
+ f g
(flg)
or
g
f
g
f
In theorem 4.1, the spectral inequality
REMARK.
LI(x,))
g
f+g+
+
+
fg
+
(fg)
J’J
if
f g
and
+
fg
+
(where
0
f, g
g
is shown to be a consequence of the well-known but seemingly unrelated spec-
-< f
f + g
tral inequality
g
+
(where
f, g
g
I
L (X,)), thus giving new proof for
the former spectral inequality.
THEOREM 4.2.
g
f, g
If
M(X,), then
I(f Ig) I
dm
:R
(-0% 0]
on
I
a
0
(fl g )dm
Ll([0,a],
m)), then
if and only if
(fg)d
I
(fg)d
X
fg
fg
X
g
(fg) e L
ja0 ( fg )dm
R
+
and
(0)
0,
it is easy to see that
l([0,a],
(fg)+
L
Ll([o,a]
m)
that the
m) whenever
implies
function
(flg)t
I
[0,), it is
([O,a], m),
-(-t)
fg)"
+(f
Since
g
+
and
:R/R
+
Since
also easy to see that
Similarly,
by noting that
is convex,
(respectively
+(fg) [(fg)+].
[(fg)+]g L l([0,a],m).
L
I ([O,a], m)
flg
(respectively
is non-constant and that both
is convex, non-negative and increasing on
and
and which is concave
0
are integrable, otherwise, there is nothing to prove
(fg)
LR+q
g
We may assume that
PROOF
R
which vanishes at
is strictly increasing such that
(flg)
(respectively
R
[0,=).
and convex on
in addition,
If
+
(4.2)
0
X
for all increasing functions
I (fg)dm
a
.< (fg)d
non-negative
-(flg)
(flg)
e
-[-(flg)
and increasing
[(6flg )+] < [(fg)+] [(6f g )+]
a
that
which
imply
[(6flg) +] dm .<
and-[_(6fg)-]..(-5[-(fg)-]-.,-5[-(flg)0
[(fg)+] dl : q[(fg) ]dm and 5[-(flg) dm .< [-(fg)-] dla : 5[-(f g
0
X
0
0
X
in
t
g
R
+
Thus, by Theorem 4 1
we have
I
dm,
whence the required inequalities follow easily.
To prove the final assertions concerning the case of equalities, let
[ X(fg)dp
152
K.M. CHONG
a
(fg)
0
for some strictly increasing function
dm
[(fg)+]
(fg)d
Then, since
potheses.
[-(f g )-]dm
0
fg)-] d
d +
(fg)
[(fg)+]
dm, and
0
+
0
[(fg)+] dm
d
X
)+]dm
[(
X
X
X
satisfying the given hy-
[-(fg)
and
a
d .<
0
[-(fg)-] dm,
[(fg)+]
case if and only if
(fg)+~ (fg) +
and only if
fg,
fg
we have equalities in the last two inequalities; this is the
[(fg)+]
and
(fg)-
-[-(fg)-],
-[-(fg)-]
and
(fg)-,
i.e., if
by Theorem 4.1 or, equivalently,
a fact which is easily seen using (2.7).
The rest follows analogously.
REMARK.
[5,
orem
(t)
When
Theorem 8.2, p.
t, Theorem 4.2 gives
102] as
Hardy-Littlewood-Plya-Luxemburg’s
Moreover, our method
an important particular case.
shows that the condition imposed earlier by Luxemburg on the functions
ifligl Ll([0,a],
e.,
The-
f
g, i.
and
not necessary for the theorem to hold, though it is a
m), is
necessary and sufficient condition to ensure the integrability of both
flg
and
f
In our version of the theorem, we have, therefore, omitted this condition, provi-
g
ded the inequality is interpreted in its appropriate sense.
A REARRANGEMENT THEOREM INVOLVING THE SUMS OF FUNCTIONS.
5.
The following extension of Corollary 3.6 also contains the inequality of Hardy-
Littlewood-Plya
[2,
Theorem 378, p.
If
f, g E
278]
[5,
and Luxemburg
102]
Theorem 8.2, p.
as
a particular case.
THEOREM 5.1.
(f + g)’-
LI(x,)
f
(f
+
g
))
$(f + g) d g
fa$(6f
+ 6
0
in addition,
(respectively
(f + g)
then
(f
for all convex functions
I
e L ([0,a]
(respectively $+ (6f + 6 g
X
If
LI(x,),
g
dm
m))
Ig)’, (f
+
:
and
I(X,)),
-
R
then the strong
(respectively
such that
+(f
more generally,
:
for all convex functions
is strictly convex such that
L
R
+ g)
$(f
+
R
6g)
/
+ g) E
+
R.
I
e L ([0,a], m)
spectral inequality
Ig)dm
153
SPECTRAL INEQUALITIES OF FUNCTIONS
(f + g)
f + g
#(f
+
6f + 6g
PROOF.
g)
(6f
(respectively
6f
(respectively
+
+
-<
ig
(f + g))
f + g)-
Ig
[i0, Theorem 3.3].
This is an easy consequence of Theorem 3.8 and
COROLLARY 5.2.
L2n(x,p),
f, g e
If
a
(6f
0
+
I
Ig) 2ndm
holds if and only if
(f + g)
is an integer, then
n ) i
where
I
2nd
X
a
(8
0
+
f
g) 2ndm.
6f
There is equality on the left (respectively on the right) if and only if
f + g
PROOF.
vex on
(t)
The result follows easily since the function
t
2n
is strictly con-
R.
COROLLARY 5 3
(Hardy-Littlewood-P61ya-Luxemburg)
2
f, g e L (X,),
if and only if
PROOF
If
6f
+
I
f, g e L (X,),
then
f6 g dm.
J0
X
If
then equality on the left (respectively on the right) occurs
L2(X,),
I
then Corollary 5.2 implies
a
(6f
0
Ig)
+
6f + 6g).
f + g
f + g (respectively
Ig
f, g e
fg d
dm
Ig
If
Ig
+ 6 ).
g
6f
f + g
(respectively
+
2
I
dm
(f +
I (6f
a
g)2
d
+ 6
0
X
2
g
dm
which, after expansions, clearly simplify to give
fa06flg Ixfg
dm
equalities’are
where the conditions for
Since
6f^t 6f
A t,
Ig
ig^t
d
If
as stated above.
t, t e R
^
am
g
/
the inequalities just proven can be
easily extended to non-negative integrable functions
f, g
by approxitions.
The
result then follows from the approximation procedure given in Theorem 3.7.
COROLLARY 5 4
tively
fg
PROOF.
f6g)
If
f, g
2
L (X,)
if and only if
6f
where
+
Ig
(X)
f + g
a <
=, then
(respectively
6flg
f + g
This is a direct consequence of Theorem 4.2 and Corollary 5.3.
(respec-
fg
6f
+ 6 ).
g
K oM. CHONG
154
REMARK.
The spectral inequality
tl p
(t)
ed by putting
+ g p
If
lf
+
p
p
g
which is obtain-
i,
in Theorem 5.1, can be used for establishing the sub-addi-
A(, p) [4]
tivity property of the norm for the Lorentz space
[9, p. 88]).
(see
Furthermore, Theorem 5.1 also contains the results obtained by Day in
6.6, pp. 942-943]
and
6.
[I, 6.4
as particular cases.
AN EXTENSION OF A THEOREM OF LONDON.
[3,
In
Theorem i and
n-tuples and, in
London proved two rearrangement inequalities involving
[i, p. 943],
[3,
It turns out that
rems.
2],
Day obtained some continuous versions of these theois also a particular case of Theorem 4.1.
Theorem 2]
Theorem i].
[3,
We shall now give a further generalization of
In either theorem, we
obtain conditions for equality for a wider class of convex functions than the one given in
[I]
or
[3]
for every strictly convex and increasing function is strictly
increasing and convex but not necessarily conversely, e.g., the identity function on
R
is strictly increasing and convex but not strictly convex.
THEOREM 6.1.
t
:
Let
ve
R.
If
R
+
+ {c
R
LI(x,)
f, g e
Let
be either both non-positive or both non-negati-
(e t)
be a function such that
[b +
+
(fg)r]r} e Ll([0,a],
m)
{c +
[b +
(flg)r]r}
(e t)
If, in addition,
r > 0, b
for some
f.<
{c +
i
(fg)r]r} <
[b +
[b
{c +
and
t g R
is strictly increasing and convex in
such that
I
(fg)r]r}
e
e I(X,)
(respectively
{c +
[b +
(fg)r]r}
L
l([0,a],
m))
b + c > O, then the left (respectively right) spectral inequality is strong if
and only if
PROOF.
flg
and
fg
(respectively
This follows directly from
with the convex function
t
(f6g)r]r}
+
i
{c + [b +
0,
0, c
I
i
then
is convex and non-decreasing in
I
(e
)
where
fg
[I0,
and
fg)
Theorem 3.3, p.
is the function
are equimeasurable.
1338]
t
/
and Theorem 3.10
log{c + (b +
e
rt r
},
R, which is easily seen to be strictly convex and increasing.
The following theorem is an extension of Theorem 3.10 for integrable functions
which are not necessarily non-negative or non-positive.
as a particular case, via
[i0, Theorem 3.3, p. 1338].
It also contains Theorem 4.1
155
SPECTRAL INEQUALITIES OF FUNCTIONS
THEOREM 6.2.
l([0,a],
+
(6f6g)+]
log+[b
+
(fg)+] e LI(x,))
(fg)+] -< log [b +
(6flg) +] 4 log [b
log [b +
b > 0
If
ely
f, g e
log+[b
log [b +
e L
Suppose
I(X,)),
and
e L
if
log [b +
m)
If
(respectively
for some
b ) 0,
(f6g)+]
(respectively
then
(fg)+]).
+
(fg)+] e L I ([0,a],
log [b +
m) (respectively
(fg)+]
then
-<
log [b +
(fg)+]
log [b +
(flg)+] "< log
(flg)
LI(x,).
+
PROOF.
log
[b +
(6fg)+]
(respectively
[b +
(fg)+])
if and only if
(fg)
+
(6fg)
+
(respectiv-
(fg)+)"
For
Theorem 4.1.
b > 0,
the result can be obtained from Theorem 3.10 and 6.1 as in
The case that
b
0
b > 0
then follows from the case that
by approx-
imations.
COROLLARY 6.3.
log
log
[b +
+
[b + (fg)-] e
LI(x,))
(fg)-] ,-<
log [b +
(fg)-]--
b > 0
f, g e
(flg)-] e Ll([0,a],
log [b +
If
ly
Suppose
+
and if
log [b +
log
log [b +
(fg)-]
e L
l(x,)),
log [b +
(fg)-]
-<
log
log [b +
(fg)-]
(fg)
PROOF.
,
[b +
log [b +
m)
If
(respectively
for some
b > 0,
(flg)-]
(respectively
log [b +
[b +
LI (X, ).
then
(fg)-]).
(flg)-]
L
1
([0,a], m)
(respectively
then
(flg)-]
(fg)-])
(respectively
if and only if
(fg)-
(flg)-
(respective-
(fg)).
This follows immediately from Theorem 6.2 on replacing
g
by
-g.
Using Theorem 6.2 and Corollary 6.3, we can give a further generalization of
[6,
Theorem l
for integrable functions which are not necessarily non-positive or non-
negative.
This generalization turns out to be of the same form as Theorem 6.1 except
K.M. CHONG
156
LI(x
g
g
f
that
)
flg,
are now any pair of integrable functions and that
(6flg) +,
can now be replaced respectively by either
and
6f6g
or
(6f6g)
7
A CONCLUDING REMARK.
(fg)
(fg)
+
and
fg
(f6g)
(6flg)
and
The particular case that
0
b
in Theorem 6.2 deserves to be highlighted as a
single theorem to be given below since it not only generalizes Theorem 3.10 but also
gives rise to all the results obtained in Sections 4 and 6 via
1338]
some
O(t)
with an appropriate convex function such as
b
0
[i0, Theorem 3.3, p.
log (b+e t)
t
ER,
for
in the case of Theorem 6.2.
THEOREM 7.1.
(respectively
log
I
f, g e L (X,)
If
+
log
are such that
+
[(6fg)+] e L 1 ([0,a], m)
I
[(fg) +]
L (X,)),
then
+- log(6f6g)
log(flg) +-<
log(fg)
log(fg)
+
(respectively
+
where the spectral inequalities are strong if
f
and
g
are either both non-negative
or both non-positive.
ACKNOWLEDGEMENTS.
The author wishes to express his sincere thanks to the referee for
his many helpful suggestions.
The results contained in this paper except for Theorem 6.2, Corollary 6.3 and
Theorem 7.1 formed a part of the author’s doctoral dissertation, Equimeasurable Rearr-
angements of Function’s with Applications to Analysis, written under the guidance of
Professor N.M. Rice and submitted to the Department of Mathematics,
Queen’s Universit
Canada in 1972.
REFERENCES
I.
Math.
DAY, P.W. Rearrangement Inequalities, Can. J.
2.
HARDY, G.H., J.E. LITTLEWOOD and G.
POLYA.
XXIV(1972), 930-943.
Inequalities, Cambridge, 1934.
SPECTRAL INEQUALITIES OF FUNCTIONS
3.
157
LONDON, D. Rearrangement Inequalities Involving Convex Functions,
Pac. J. Math. 34(1970), 749-753.
4.
LORENTZ, G.G. and T. SHIMOGAKI.
Interpolation Theorems for Operators in
Function Spaces, J. Func. Analysis 2(1968), 31-51.
5.
LUXEMBURG, W.A.J. Rearrangement Invariant Banach Function Spaces, Queen’ s
Papers in Pure and Applied Mathematics, 10(1967), 83-144.
6.
HEWITT, E. and K. STROMBERG. Real and Abstract Analysis, Springer-Verlag,
New York, 1965.
7.
CHONG, K.M. Spectral Orders, Uniform Integrability and Lebesgue’s Dominated
Convergence Theorem, Trans. Amer. Math. Soc. 191(1974), 395-404.
8.
HARDY, G.H., J.E. LITTLEWOOD and G. POLYA.
by Convex Functions, Mess.
9.
Some Simple Inequalities Satisfied
of Math. 58(1929),
145-152.
CHONG, K.M. and N.M. RICE. Equimeasurable Rearrangements
of Functions, Queen’s
Papers in Pure and Applied Mathematics, No. 28, 1971.
i0.
CHONG, K.M.
Some Extensions of a Theorem of Hardy, Littlewood and
Their Applications, Can. J. Math. XXVI(1974), 1321-1340.
Polya
and