BULL. AUSTRAL. MATH. SOC.
00A99, 35J65
VOL. 50 (1994) [317-326]
ON SOME INEQUALITIES FOR
ELEMENTARY SYMMETRIC FUNCTIONS
Mi LIN AND NEIL S. TRUDINGER
In this note, we prove certain inequalities for elementary symmetric funtions that
are relevant to the study of partial differential equations associated with curvature
problems.
In this note, we prove certain inequalities for elementary symmetric funtions that
are relevant to the study of partial differential equations associated with curvature
problems, (see, for example, [2, 3, 7]). In particular our first theorem relates to the
partial uniform ellipticity of the higher order mean curvature operators while our second
one is an improvement of an inequality of Ivochkina that was crucial in her study of
these operators in [2, 3]. From these two inequalities, we deduce further inequalities
that arose in our treatment of curvature quotients in [5].
We begin with some definitions and notation. First the fc-th order elementary
symmetric function of n variables, Sj,, is denned by
(i) sk(\)=
where 1 ^ A: < n and A = (Ai,... , An) € R n . For consistency, we extend 5* by setting
50(A) = 1,
=0 for k > n.
The function 5* will be considered in the corresponding cone in R n , Tk, given by
(2) Tk = {A G R" | Sj(\) > 0, £arallj = l , . . . , * } .
It is easily seen that P* is , in fact, a cone with vertex at the origin. Clearly F* C Tj
for k ^ j and Tn is the positive cone {A 6 R n | A,- > 0, i = 1,... , n } . For any fixed
t-tuple {ii,i2,--- ,*t} C {1,2,... , n } , we define
(3) •S*!i1i,...i,(A)
Received 23rd December, 1993
Research partially supported by Australian Research Council Grant.
Copyright Clearance Centre, Inc. Serial-fee code: 0004-9729/94 SA2.00+0.00.
317
318 M. Lin and N.S. Trudinger [2]
that is, Sjtjt!...,-, is the fc-th order elementary symmetric function of the n — t variables
{!,-•• ,™}\{*i.--- ,**}•
The following properties of the functions 5* will be used in this paper:
(4) Sh(X) = Skii(\) + A;Sfc_1;i(A)
(5)
t=i
for all A G R n . Furthermore, if A G F t , then at least k of the numbers A i , . . . , An are
positive and moreover
(6) %!*,...<.(*) > 0
for all { i i , i 2 , . . . ,i,} C { 1 , 2 , . . . , n } , l + s ^ fc. As well, we have the Newton inequal-
ities
for A G R n , k ~£ 2 and the Maclaurin inequalities
ri V'k
(S) []
for A G T i , A: > / > 1; (see [6]).
Throughout this paper we shall write A in decreasing order,
(9) Aj > • • > \ k > • • • 2 AP > 0 ^ A p + 1 • • • > An
where p ( ^ &) is the number of positive Aj.
Our first theorem provides lower bounds for the ratios Sk-v,i/St-i.
THEOREM 1 . There exists a positive constant 8, depending on n and k, such
that
for aU O *, A G r fc .
[3] Elementary symmetric functions 319
REMARKS, (i) It suffices to prove (10) for the case % = k, because Sk-v,i ^ Sk-i,j if
i ^ j . This follows from the formula,
and the positivity of Sk-2;ij on F t .
(ii) Our proof will yield the following estimate for 6, namely
k k
3=2 i=j
In the special case n = 3, k = 2, we obtain 8 ^ 1/3, which is sharp, as is evidenced by
the example Ai = A2 = 1, A3 = —1/2. We provide an example later to show that the
condition i ^ k cannot be improved.
(iii) When i > p, that is A; ^ 0, the estimate (10) is already known, in conjunction
with gradient bounds for curvature equations [1, 4]. In this case, (10) follows, with
0 = 1, immediately from the formula,
•Sjb-l(A) = Sk-l;i + Aj5fc_2;i.
PROOF OF THEOREM 1: Wefirstprove the inequality,
(12) ISi-ljlfcl < CkSk-l;k,
Using the formula (4), we have
Sk;lk + ^lSk-l;lk ~ •!
(13) >
(14) Sk-i;ik + Ai5t_2iijb = Sfc-u
Eliminating Ai from (13) and (14), yields
^ Sk-l;k[Sk-l;lk
= Sk-l,kSk-l;l
so that by Newton's inequality (7) we obtain
320 M. Lin and N.S. Trudinger [4]
whence (12) follows.
Now from (12) and (14), we have
so that
S > ^
Let us now suppose that (10) is valid wherever k and n are replaced by k — 1 and
n — 1, that is for some positive constant 0 — 0{k — l,n — 1), we have
Sk-2;\k
Here we axe replacing 5* by Sk-i-i • Then we obtain
sk-lik > T+
whence we conclude
(15)
Since inequality (10) clearly holds for k = 1, with 0 = 1, we are done. D
FURTHER EXAMPLES
(i) Taking k < n and Ax = M, A< = 1 for 1 < i < k, Xk = M " 1 , A; = 0 for i > k,
we clearly have
Sk-l;k-l _ 1
5fc_! M + (k - 2) + M"1
—» 0 as M —> co
which shows Theorem 1 is impossible for i < k.
(ii) For * = 2 and n ^ 3 , we get from (11),
0(n,2)>-
- 2)/(n -
[5] Elementary symmetric functions 321
As with the case n = 3, 0(3,2) = 1/3, we also must have equality in this estimate.
This follows from the example
A< = 1 for i = 1 , . . . ,fc,
Xi = a for i = k + 1 , . . . , n
where
Accordingly we have the sharp estimate
(16)
Si 2 + y/2(n - 2)/(n - 1)
1
2 + \/2
The last inequality also follows directly from the formula
(17) 0<2S2 = S\- t=i
Note when 0 is sharp, equality in (10) must be attained on dTk, otherwise the k
partial derivative of the ratio must vanish, that is
which contradicts A £ F t according to property (6).
Our second theorem is an improvement of the Ivochkina inequality [2].
THEOREM 2 . There exists a constant C depending on n and k such that
(18) 5 4+1;r (A)< £ 5*_1;i(A)(A,)2
for all A e Tk.
PROOF: We separately estimate each of the terms Q in the sum of the left hand
side of (18). If Q has an odd number of negative Aj, we are clearly done. Let us assume
first that Q has an even number of negative factors and write Q in the form
Q = AXi\j Xi,Xj<0
322 M. Lin and N.S. Trudinger [6]
and
A — \. ... \.
x
**• — *1 k —1 '
Without loss of generality, we can order A,
K > • • • > A,-. > 0 > Ai<+1 ^ Xik_x.
Consequently
< A ...A- lA- I*"'-1
Since A G Tk, the sum of any n — k + 1 of the A< is positive and hence
and thus
A < (n - k + 1)*~—2AiA2 • • • A.(At)*-'- 1
^ (n — k + 1) Ai • • • A,A,+i • • • Xk-i-
Next by expanding
= Sk-l;l + Al5fc_2;12 + h Al • • • Afc_2Slil2...(jfc_l) + Al • • • Afc-J
we have the inequality
(19) AiAa-.-Ai-^Si-i.
Hence
2
^ ^ ^ - ^ ^ - ^ ^ [ ( ^ ^ ( A , ) ]
^ \{n - k + l ) * - 1 ^ - ! . ^ ) 2 + S4_1.,-(A,-)2]
by virtue of (10) with 6 = 1, Remark (iii). The case when Q has all positive factors
follows directly from (19) and Theorem 1, although it could be deduced independently
of Theorem 1. D
[7] Elementary symmetric functions 323
REMARK. We have dispensed with an assumption made by Ivochkina [2, condition
(2.7)] and our summation is taken from it t o n rather than 2 to n . The latter improve-
ment is essential for our applications to curvature quotients.
APPLICATION. Observing that
PO) « * . 5l_1,i,
we may write the estimates (10) and (18) in the form
(22) Sk+lir ^ C V |^(A,)2 for A € Tk, Ar > 0.
We need an extension of (22) for quotients of elementary symmetric functions
(23) Sh,i = § • for ft > I Z 1,
which follows by combination of Theorems 1 and 2.
THEOREM 3 . There exists a positive constant C depending on n and k such
that
for all A G Tk.
PROOF: We calculate, (as in [7]),
Sf
sf
Tl(k — I) Sl-fiSk—l;i
(by Newton's inequality)
k(n - I) Sf
for i ^ / + 1, by Theorem 1. The desired inequality (24) then follows directly from
Theorem 2. D
324 M. Lin and N.S. Trudinger [8]
Next, following Ivochkina [3], we estimate
{
(26) ^ (\r - i s l s r ) Sk + Cj2 St_1;iA?
by Theorem 2, where C is a further constant depending on Jb and n. Consequently we
have
(27, M ^
with S\ replaceable by Ar, when k~^2. For quotients, we now prove a stronger version
of (27).
THEOREM 4 . There exists a positive constant C depending on n and k such
that for 1^1, A G r fc
PROOF: Let us write
(29)
From (25) we have
(30)
by (4). To estimate the second term on the right hand side of (29) for i = r, we sum
(29) over i ^ r, to obtain
(31) $ > A ? = | [ ( Z + 1)5*1 - S,rXr] -
[9] Elementary symmetric functions 325
so that by Theorem 3,
(32) _^A r < i[l + (Jb + 1)C] J ^ A ? + -£g[Si;r\r - (i + l)Si+i]
where C is the constant in (24).
Since Sj+i(A) > 0, it remains to estimate the term
F
JtSl'rXr
on the right hand side of (31). First we observe from the proof of Theorem 2, that any
term Q in Sj;rAr can be estimated by
l+1
(33)
so that by (19),
(34) s
Similarly we can prove the estimate
(35)
for any j ^ k. For, using
Sk = Sktj + \jSk-i,j,
we see that (35) is automatically true if Sk-,j ^ 0. Otherwise we estimate as above
by (19). Similarly to the above estimation, we also have, for j < / + 1,
A,+iS,<
(36) ^
Combining (34), (35) and (36), we thus obtain for j < / + 1,
hr r
S, *° 5? *
(37) ^ CFj\)
where C depends on n , k,l. By choosing j ^ r , we thus complete the proof. D
REMARK. Theorem 4 is a crucial inequality in our derivation of second derivative esti-
mates for solutions of prescribed curvature quotient equations [5].
326 M. Lin and N.S. Trudinger [10]
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[1] S. Caffarelli, L. Nirenberg and J. Spruck, 'Nonlinear second-order elliptic equations V.
The Dirichlet problem for Weingarten hypersurfaces', Comm. Pure Appl. Math. 41 (1988),
47-70.
[2] N.M. Ivochkina, 'Solution of the Dirichlet problem for curvature equations of order
m', (English translation), Math USSR Sb. 87 (1990), 317-339.
[3] N.M. Ivochkina, 'The Dirichlet problem for the equations of curvature of order
m', (English translation), Leningrad Math J. 2 (1991), 631-654.
[4] N.J. Korevaar, 'A priori interior gradient bounds for elliptic Weingarten equations', Ann.
Inst. H. Poincare, Anal. Non Lineairel (1987), 405-421.
[5] M. Lin and N.S. Trudinger, 'The Dirichlet problem for the prescribed curvature quotient
equations', Topological Methods in Nonlinear Analysis 3 (1994).
[6] D.S. Mitrinovic, Analytic inequalities (Springer-Verlag, Berlin, Heidelberg, New York,
1970).
[7] N.S. Trudinger, 'The Dirichlet problem for the prescribed curvature equations', Arch.
Rational Mech. Anal. I l l (1990), 153-179.
Department of Mathematics Centre foi Mathematics and its Applications
Northwestern University The Australian National University
Evanston IL 60208-2730 Canberra ACT 0200
United States of America Australia
