49
SOME INEQUALITIES CONCERNING SYMMETRIC FORMS
J. N. WHITELEY
1. Let av ..., am and bv ..., bm be non-negative real numbers.
well-known inequality of Minkowski states that
{2 (a,+6,)»}V» < {£ a,»}1/re+ {2 bffln
The
(1)
if n ^ 1. If n is a positive integer, this inequality asserts a property of a
particular symmetric form (i.e. homogeneous polynomial) in m variables,
namely the sum of the n-th powers of the variables. Some time ago,
Prof. A. C. Aitken conjectured that similar properties are possessed by
certain other symmetric forms. In particular, let EW)(a) denote the
w-th elementary symmetric function of av ..., am and let C-n)(a) denote
the «-th complete symmetric function of av ..., am, the formal definitions
being
..+»,„ = «
0or 1
..a£,
(2)
(3)
Then Prof. Aitken conjectured that
,
(4)
.
(5)
The inequality (4) has been proved by H. F. Bohnenblust (unpublished)
and by M. Marcus and L. Lopes*.
The object of the present paper is to exhibit both (4) and (5) as
particular cases of an inequality concerning a class of symmetric forms
containing a real parameter K. The inequality (4) will be the case K=1,
the inequality (5) will be the case K= — 1, and Minkowski's inequality
(1) will be the limiting case as /c-s-0 (K < 0).
The general symmetric form of the class in question is T^(alt
am),
defined by
(6)
where
if « < 0 .
(8)
* Canadian J. of Math., 9 (1957), 305-312.
[MATHBMATIKA 5 (1958), 49-57]
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50
J. N. WHITELEY
Thus JW is homogeneous of degree n in av ..., am. Its coefficients are
non-negative provided that, if K is positive but not an integer, we impose
the restriction n<K-\-l.
The definition can be expressed alternatively
by a generating function:
(9)
n=0
with ai replaced by —aj if K < 0.
We prove:
THEOREM.
If K <0, we have
{T^ia+b)}1'"- < {TW(a)yin+ {T^{b)f'n.
(10)
{T<-n\a+b)ytn > {TW(«)}1/re+ {TW(b)yin,
(11)
/ / K > 0, we have
provided that W < K - J - 1 &/ /C is no£ an integer.
If K = 1, (7) gives X{ = 1 for i — 0 or 1 and Ai = 0 for i > 1, so that
£><„)_ if K = _ i ( (g) gives Ai = 1 for all i, so that TW = C<-n\ If
K = — 8, where S is a small positive number, we find that
T(n) =
m
T^>{a) = n~1h 2 a
i=i
so that Minkowski's inequality follows from (10) on making 8->0.
The proof uses the classical theory of maxima and minima, together
with induction on m and n. A supplementary argument is needed when
K is a positive integer, since then the extremal problem relates to an
unbounded region.
The inequalities of the theorem are equivalent to the assertion that
the surface defined by T(n) (xv ..., xm) = 1 in the positive " quadrant "
of m dimensional space is convex if K < 0 and concave if K > 0. I have
not been able to find a proof which is directly related to this interpretation.
The inequalities can be extended to the asymmetric forms obtained
by replacing K by m parameters KV ..., Km, provided these are all of the
same sign.
Another approach to the inequalities (10) and (11), which was suggested
by Prof. Aitken himself, is by means of polar forms. If f(x) is a form of
degree n in m variables, the coefficient of V in f(ax-\-tbx, ..., am-\-tbm) is
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SOME INEQUALITIES CONCERNING SYMMETRIC FORMS.
51
the polar form
H^Ja,
b) =
S
(i1l...ij)-1b^...bi"
•i1+...+im=v
d
"
In particular, Hn0=f(a) and HOn=f(b).
for the validity of the inequality
Hence a sufficient condition
Hn_VtV {a, b) < ( ^HtnH1-^
is that
J(a).
oa^ ... oam
(12)
for v = 1, ..., n— 1. It is possible to prove, by the methods of this paper,
that (12) holds if / = T w and K < 0, and that (12) with the inequality
reversed holds if / = T<n) and K > 0, with the same proviso as in the
theorem.
2. We first make a general remark concerning symmetric forms. Let
fm(av ..., am) be a symmetric form of given degree in m variables, defined
for all m. Such a form may or may not have the character of " consistency "
expressed by the identity
/ m - l K . •••> V l ) =fm(aV
•••> am-l> °)'
(12a)
It is important for the proof that the form TWfa, ..., am) does have this
property.
Let the operator #, applicable to forms in m variables alt ..., am, be
defined by
m
&= S (d/daj).
(13)
3=1
The effect of & on T<w)(a) is easily evaluated, as follows.
LEMMA 1.
Proo/.
We have*
&TW(a) = (—m/c+»—1) T<»-«(a) /or K < 0,
(14)
&T<n\a) = (mK—n+1) T<n-«(a) /or K > 0.
(15)
By (6),
S
* It will be noted that, as a consequence of these results, the form dTi") satisfies an
identity similar to (12a) but with a numerical factor independent of o lf ...,am. This
property characterizes the forms T^(a), considered in the present paper, among a wider
class of symmetric forms.
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52
J . N. WHITELEY
If K < 0, we have, by (8),
(*+l)A, +1 =(i-K)A<-
Hence
(dldat)T^=
S
(»,-«) A,,.--**.'**1 •••«»•
(16)
i 1 +...+fm=Jl-l
Summation over j from 1 to m gives (14), since
m
2 (ij—K) = n— 1—m,K.
1=1
The case K > 0 is similar, with K—i in place of i—K.
We require also a result, in the form of an inequality, for the effect
of an operator in which the summation over j in (13) is incomplete.
LEMMA
2. Suppose that 1 ^ s ^ m — 1 .
Then, if
K<
0, we have
if K > 0 we have
2 (d/3a j )T<">(a)>((m-s)K-«,+
l')27<n-1>(a),
X
3=S+1
(18)
'
provided that n < K + 1 if K is not an integer.
Proof. Suppose first that K < 0. Then (16) gives
Hence
S
3=1
whence
S
3=S+1
= (—(m—S)K+»—
by (14). The case K > 0 is similar; the condition «, < K + 1 if /c is not an
integer ensures that Xt > 0 for all i concerned.
3. Proof of the Theorem for K < 0. We use a double induction on
m and n. The inequality (10) holds (with equality) for all m if n= 1,
and for all n if m = 1. We shall prove that it holds for a particular pair
m, n (m^2, n^2) provided it holds for all pairs m', n with m' < m and
all pairs m, n' with n' < fi.
Let ax, ..-, am, bx, ...,bm be variables which are subject to the
conditions
TW(a1+b1,...,am+bJ=l,
(19)
, ...,om>0,
(20)
&!><), . . . , & m > 0 .
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SOME INEQUALITIES CONCERNING SYMMETRIC FORMS.
53
These conditions define a closed set of points in 2m dimensional space.
This set of points is also bounded, for f"'^, ..., xm) has non-negative
coefficients and includes the terms
where*:' = —«:> 0,sothat (19) implies upper bounds for ax-{-&!, ..., am-\-bm.
Let M denote the minimum of
{TW(a)}v»4-{yw(6)}i;»
(21)
subject to the above conditions. It suffices to prove that M ^ 1, since
this implies (10) by considerations of homogeneity.
Suppose first that the minimum is attained at a point for which
o 1 > 0 , ...,am>0,
b1>0,...,bm>0.
This point cannot be a singular point on the surface (19), for by Euler's
theorem on homogeneous functions we have
'Lbi£- T<n\a+b) = nT^(a+b) = n,
(22)
so that the first partial derivatives cannot all vanish. Hence Lagrange's
relations for a local extremal of (21) subject to (19) are applicable. They
are:
A {T()(a)}V_A
{T(»)()}V»A A {TW
{TW(a+b)yi
= 0,
A {T<n)(&)}iyn_A A {TW(a+&)}i/» = o,
where A is an undetermined multiplier.
yw( ffi )
=
Hence
A { 2 ( ) ( a + 6 ) } + / ^ T<n\a+b),
A TW(b) = \{T^\a^-b)Yx^n~T{-n\a^b).
00j
(23)
(24)
00j
We combine these relations in two different ways. First we multiply
(23) by at and (24) by bjt and sum for i and j from 1 to m. By Euler's
theorem on homogeneous functions, we obtain
.
By (19) and the definition of M, this implies \ = M.
(25)
Secondly, we sum
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54
J. N. WHITELEY
over i in (23) and use the result of Lemma 1, in the two forms
where C depends only onTO,n, K. We obtain
{TW(«)}-w/» r<n-i)(o) = A{T<»>(a+6)}-1+1'm r<»-»(a+&).
(26)
Similarly
(27)
Raising both these equations to the power l/(n—1), and adding, and
using (25), we obtain
- 1 ).
(28)
But by the inductive hypothesis, with n' = n—l, we have
-«.
(29)
Hence A ^ 1, that is, M^l.
Suppose next that the minimum of (21) is attained at a point at which
one or more of av ..., am, bv ..., bm is 0. We can suppose that at and bt
are not both 0 for any i, for in that case the result would follow from the
inductive hypothesis withTO'<TO,in view of the property of consistency
remarked upon at the beginning of §2. Thus without loss of generality
we can suppose the minimal point has
ai
= . . . = a g = 0, br+1=... = bm = O,
(30)
where q ^ r, and has all the other a^ and &,- positive.
The minimum M is also the minimum of the function
8+
(bv ..., br)}^
(31)
of TO—q-\-r variables, subject to
T^{bx, ..., bv aq+1+ba+v ..., ar+1, ..., am) = 1,
(32)
with all the variables non-negative. These conditions define a closed
and bounded set of points in a space of m—q-\-r dimensions. The minimal
point is again a non-singular point on the surface (32), for the relation
(22) remains valid if i is summed from q-\-1 toTOand j is summed from 1 to r.
Hence Lagrange's conditions for an extremal of (31) subject to (32)
are applicable. They again give (23) and (24), except that i and j are
limited to the ranges i > q and j ^ r. The deduction (25) remains valid,
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SOME INEQUALITIES CONCERNING SYMMETRIC FORMS.
55
on the understanding that a denotes the set aq+v ..., am and b denotes the
set bv ..., br. We again have A = M.
Summation over i from g+1 to m in (23) gives
S J-TW
£
=q+ioai
where !T<w)(a+&) now denotes the function on the left of (32). By Lemma 1
applied to T(n)(«s+i> •••, «m)> w e n a v e
i=q+X°ai
By Lemma 2 applied to the function on the left of (32), we have
— (m—q)K-\-n—'.
Hence
This corresponds to (26), but with inequality in place of equality. Similarly
the inequality analogous to (27) holds. Combining these as before, we
obtain (28) with the sign ^ , and this suffices for the deduction that
M = \ ^ l , using again the inductive hypothesis with n' = n—1.
4. Proof of the Theorem for K > 0 and not an integer. The proof of §3
applies, with the signs of inequality reversed throughout, and with
maximum instead of minimum. Note that, by the condition n<.K-\-\,
the form TM (xv ..., xm) contains the terms
and consequently the region defined by (19) and (20) is again bounded.
Note that the condition n < K-\- 1, imposed in the present case, prevents
one from deducing the result in the next case by considerations of continuity.
5. Proof of the Theorem for K a positive integer. It suffices to prove the
result when K = 1, for it is clear from (9) that the general case follows on
considering the set of m,K numbers obtained from the set av ..., amhy
repetition K times. When K = 1 we have T<™> = EM, and the inequality
to be proved is (4). We again proceed by induction on m and n.
Let P be a large positive number, and let av ..., am, bv ..., bm be
variables which are subject to the conditions
^<»>K+&i> ...,am+bj
= l,
(33)
P, ..., 0 < 6 m < P .
(34)
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56
J. N. WHITELBY
Let M denote the maximum of
(35)
subject to the above conditions. We have to prove that M ^ 1 .
If the maximum is attained at a point at which ai < P and bt < P for
all i, the proof of §3 (with the signs of inequality reversed, as in §4) is
applicable. Thus it remains only to consider the possibility that the
maximum is attained at a point for which ax = P (say).
We have the identity
™(x2, ..., xm).
(36)
By this identity with x = a-\-b, and (33),
E("-V(a2+b2, ...,
am+bm)^P-\
By an inequality of Maclaurin*,
where G depends only on m and n.
EW(az+b2,
Hence
...,an+bm)<e,
where e = e(P)->0 as P->oo. A fortiori, we have
#<">(a2, ..., am) < e, EO*(bt, ..., bm) < e.
By the identity (36) with x = a,
Writing
A = E<n-V(a2, ..., am)
for brevity, we have
Similarly with b and B for a and A.
Hence
M<(a1A)1'n+(b1B)1ln+2e1ln.
(37)
By Holder's inequality,
(O1A)V»+ {bx B)Vn < (o1 + 61)V»{^1'(n-«+^(n-DJCn-W*.
(38)
By the inductive hypothesis, with n' — n—l, we have
Hardy, Littlewood and P61ya, Inequalities (1st ed.), Theorem 52.
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SOME INEQUALITIES CONCERNING SYMMETRIC FORMS.
57
The right-hand side does not exceed
by (36) with x = a+b.
Hence, by (33),
^i/(»-i>+£i/(»-i) ^ (a1+b1)-1^n-1K
It now follows from (37) and (38) that
Since M obviously increases with P, it follows that M ^ 1, and this
completes the proof.
I am grateful to Prof. Davenport for telling me of Prof. Aitken's conjectures and for preparing this paper for publication.
Department of Mathematics,
University College,
London.
(Received 1st October, 1957.)
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