Taylor & Francis
J oflnequal. & Appl., 2002, Vol. 7(4), pp. 603-611
Taylor & Francis Group
The Inequality of Milne and its
Converse
HORST ALZER a and ALEXANDER KOVA(EC b*
aMorsbacher Str. 10, 51545 WaldbrSI, Germany; b Departamento de
Matemtica, Universidade de Coimbra, 3001-454 Coimbra, Portugal
We prove: Let wj > O(j
,n;n > 2) be real numbers with )-’’’=l wj
for all real numbers pj E [0, 1)(j 1,... ,n):
1. Then we have
with the best possible exponents
and/Y 2 minl<A<_n wj. The left-hand side of (0.1)
with
is a discrete version of an integral inequality due to E.A. Milne [1 ]. Moreover,
we present a matrix analogue of (0.1).
Keywords: Milne’s inequality; Inequalities for sums; Matrix inequalities
2000 Mathematics Subject Classification 26D15, 15A45
I INTRODUCTION
A problem in astrophysics led E. A. Milne 1 in 1925 to the following
interesting integral inequality:
f(x)g(x)
dx
f(x) + g(x)
If(x) + g(x)]dx <
f(x)dx
g(x)dx. (1.1)
*Correspondence author. E-mail: [email protected]
ISSN 1025-5834 print; ISSN 1029-242X. (C) 2002 Taylor & Francis Ltd
DOI: 10.1080/1025583021000022405
H. ALZER AND A. KOVA(EC
604
This inequality holds for all functions f and g, which are positive and
integrable on [a, b]. In view of
f(x) + g(x)
(1.2) yield a refinement of the classical
Cauchy-Schwarz in-equality for integrals. A discrete version of (1.1)
and various related results are given in the monograph [2, pp. 61-62].
In 2000, C.R. Rao [3] combined (1.1) and the arithmetic mean-geometric mean inequality to obtain a very elegant double-inequality for
we conclude that (1.1) and
sums.
PROPOSITION
j=l wj 1.
Let
wj
Then
> 0(j-- 1,...,n;n >_ 2) be real numbers
we
have
for
all
real
numbers
with
pj E
(-1, 1)(j- 1,...,n):
Rao applied methods from statistics to provide a short proof of (1.3). A
statistical interpretation of (1.3) is given in [4]. We remark that the lefthand side of (1.3) is a special case of the well-known Tchebyschef inequality; see [2, p. 43].
The double-inequality (1.3) admits a matrix version in the sense of
positive semidefinite (psd)- ordering for Hermitian matrices. An extensive treatment of positive definiteness can be found in [5, chapter 7].
The left-hand side of the following matrix inequality is due to Rao
[3]. Using the ideas he indicated on the last lines of page 313, we conclude that the right-hand side holds, too.
THE INEQUALITY OF MILNE AND ITS CONVERSE
605
PROPOSITION 2 Let wj > 0(j 1,... ,n;n >_ 2) be real numbers with
j wj 1. Then we have for all Hermitian commuting matrices
A, V,..., Vn with A > O, -A < Vj < A j 1,...,n)"
n
E
j=l
n
n
j=l
j=l
(1.4)
j--1
It is natural to ask for improvements of (1.3) and (1.4). In Section 2 we
establish: if pj [0, 1)(j 1
n), then in the second inequality of
(1.3) we can replace the outer exponent 2 by the smaller term
2- min<<, w. And, in Section 3 we present a corresponding result
for matrices, which leads to a sharpening of the fight-hand side of (1.4).
2 DISCRETE INEQUALITIES
The following theorem presents a refinement of the second inequality of
(1.3) and shows that the first inequality is sharp in a certain sense.
THEOREM
)"=,
wj
Let wj > 0(j 1,...,n;n _> 2) be real numbers with
1. Then we have forall real numberspj E [0, 1)(j 1,... ,n)"
,
wj
j=
<
-.=
1-
wj
,
wj
1-pj .= l+pj
j--1
1--Ps2
(2.1)
with the best possible exponents
and
fl=2-
min wj.
<_<_< <_n
Proof To prove the second inequality of (2.1) with fl
we may assume that
0
<Pn
<_Pn-1 <
<_P2 <Pl < 1.
2- min<A_<n wj,
H. ALZER AND A.
606
Let w
KOVAtEC
mini <_<n wj and
j=l
j=l
Further, we define for q
6
{1
and p
n-
fq(P) f(P
6
[0, 1):
Pn).
P, Pq+I
Let Pq+l < P < 1. Differentiation gives
WqZ
fq’(P) (1 p2)ABC’
(2.2)
where
q
mq Z
j=
_p2
n
wj,
A
mq-Jr
Z
j=q+
wj
2,
B=
p)
mq.--
n
j=q+
"
C= Wq + E
l+p
+pj
j=q+l
and
Z
Since 0 < Pn <
>
A[(1 -p)B
+ p)C] + 2(2
w)pBC.
(2.3)
< Pq+l < P < 1, we have
-p >...>
1-pq+l
(1
-p
1-Pn
and
<
l+p
+ Pq+l
l+p
l+pn
THE INEQUALITY OF MILNE AND ITS CONVERSE
607
so that Tchebyschef’s inequality gives A < BC. Hence, (2.3) leads to
Z
> (1 -p)B
(1 + p)C + 2(2
p)2 (1 + p)2
n
pj
j=q+l
> 2p[2
w)p
w
mq] +
,
1
n
2p[2
Wq]
w
)
wj[(l p)2
(l
+ p)2
j=q+l
2p(Wq
w) > O.
(2.4)
From (2.2) and (2.4) we obtain that fq’(p) > 0, which implies that fq is
strictly increasing on [pq+, 1). Hence, we get
f(P
P,O =Ji(P) >j](P2)=J(P2) >J(P3) >"" > fn-l(Pn-)
>" L-1 (Pn)
--(1 W) log(1 p2n) O.
(2.5)
_.
This proves the second inequality of (2.1) with fl 2- min<_<n wj.
Moreover, iff(p,... ,Pn)= 0, then the strict monotonicity offq implies pl--’"=Pn and the fight-hand side of (2.5) yields p 0.
Thus, we obtain f(pl
Pn) >_ O, with equality holding if and only
-
ifp
=Pn O.
It remains to show that in (2.1) the exponents a
and
for
all
is
We
that
valid
are
2sharp.
suppose
(2.1)
min<A_<n wj
fl
n}.
n). Let mini <<_n wj wk with k 6
pj [0, 1)(j
We set pj 0 for j
n,j k, and Pk > 0. Then the second inequality of (2.1) leads to
+ 1 ’/
or, equivalently,
log(1
<
+
+ wkpk/(1 -pk)) + log(1 wkpk/(1 +pk)) < ft.
log(1 + wkp2/(1 _p2,))
(2.6)
Applying l’Hospital’s role we conclude that the ratio on the left-hand
side of (2.6) tends to 2 wk, ifpk tends to 0. Thus, we get/3 > 2 wk.
If we set p
Pn (0, 1), then the first inequality of (2.1) gives
(1/(1 -p)) < 1/(1 -p), which implies < 1.
H. ALZER AND A.
608
KOVAEC
Remarks (1) The sign of equality holds in the first inequality of (2.1) with
if and only ifp
Pn. The proof of Theorem reveals that
equality is valid in the second inequality of (2.1) with fl=
2 min<_<, wj, if and only ifp
--Pn O.
n), then the left-hand side of (2.1) holds
(2) If IP#I < l(j
with the best possible exponent
1. This follows from Tchebyschef’s
if
inequality. Further, IPjl < l(j
n), then the right-hand side of
(2.1) is valid with the best possible exponent fl 2. Indeed, if we set
p -p2 0,p3
p,, 0, then (2.1) gives
log(wp/(1-p)-w2p/(1 +p) + 1)
+ log(w2p/(1 Pl) wpl/(1
log((w, + w2)p/(1--p)-" 1)
We let p tend to
and obtain
+ p ) + 1) <_ ft.
fl > 2.
(3) The following integral analogue of Theorem holds:
o
Let w be a positive and integrable function on [0, 1] with
w(x)dx 1. For all integrable functions p’[0, 1] --+ [0, 1) we have
w(x)
(p(x)) 2
dx
<
<
with
*
and fl*
p(x)
dx
(p(x)):
o
4- p(x)
dx
dx
2.
Moreover, these constants are sharp. In fact, if we set p(x)=- 1/2,
then the left-hand side gives (4/3)* < 4/3, which implies *< 1.
And, if 6(0,1) and p(x)=l/2 for O<_x<f,p(x)=O for
6 < x < 1, then we get
log(1
f06
+ I(i)) + log(1 1(6)/3) < fl*,
log(1 + 1(6)/3)
where 1(6)
w(x)dx. We let 6 tend to 0 and get fl* > 2. Hence, on
contrary to the discrete case it is not possible to replace the exponent
fl* 2 by a constant, which is smaller than 2.
THE INEQUALITY OF MILNE AND ITS CONVERSE
609
3 MATRIX INEQUALITIES
Our second theorem provides a striking companion of Theorem 1 for
matrices. For the reader, who is unfamiliar with this area, the proof contains some indications of how to interpret the result, but he may have to
become more familiar with the subject to see the steps of the proof fully
justified.
1,...,n;n >_ 2) be real numbers with
wj > 0(j
and let I be the unit matrix. Then we have for a commuting
family of (Hermitian) matrices P1,.. ,Pn with 0 <_ P) < I(j 1,... ,n):
THEOREM 2 Let
jn=
wj
(3.1)
with the best possible exponents
a
l
and b=2- min
l<_j_<n
Proof We assume the matrices of size m x m. In the face of (1.4) (with
A
I) it suffices to prove the second inequality of (3.1) with
b 2 mini <A<, wj. We conclude from our assumption on the P that
there exist a unitary matrix U and diagonal matrices Aj such that
UU*= Aj diag(2j, ...,2m); see [5, p. 172]. Up to rearrangements of the diagonal entries, the diagonalization is unique and the
m) represent the eigenvalues. By definition and character2(i =
ization of positive semidefiniteness we obtain that 0 < 20. < 1 for all
andj.
We recall that iff is a function defined on all eigenvalues of a diagonalizable matrix, say L SAS -1, with A diag(2
2,), then we
define f(L)= S(f(21)
f(2,))S By the ’functional calculus’ for
matrices, [6, p. 408], we know that f(L) depends neither on the diagonalizing matrix nor on the original order of the diagonal elements. This
definition is conservative: if applied to traditional matrix functions like
-.
H. ALZER AND A.
610
KOVAtEC
taking integer powers, then it agrees with the usual definitions; similarly,
in this case, a diagonalizing matrix is given by S.
From these observations it is clear how to interpret the matrix analogue of the fight-hand side of (2.1). We find that this analogue is equivalent to
diag
]
j
wj
diag
1-20.’
j
+ 20
w
1-2
< diag
(3.2)
where we have presented the/-th diagonal entry of each matrix. Now, as
an immediate consequence of the second inequality of (2.1) we conelude that (3.2) holds with b 2 min <<_n wj. It is clear from Theorem that this exponent is best possible.
Remarks (1) The sign of equality holds in the fight-hand side of (3.1) if
and only if it holds componentwise in the diagonalization; according to remark (1) after Theorem 1, this is the case if and only if we have Pj 0 for
all j; the absence of this condition does not mean however that the inequality in (3.1) is strict. The sign of equality is valid in the first inequality of
(3.1) if and only if P
Pn.
-
(2) The substitution Pj ViA and a simple calculation lead to the following refinement of the fight-hand side of (1.4):
Y
j=l
wj(A
Vj) -I
Y
j=l
wj(A + Vj) -I _< A 2b-2
wj(A 2
Vj2) -I
=1
where A, V
V, is a commuting family of Hermitian matrices satis<
fying 0 Vj < A(j
n) are positive real
n),wj(j
numbers with
1, and b 2- min<_<,, wj.
wj
Y4I
References
[1] E.A. Milne, Note on Rosseland’s integral for the stellar absorption coefficient, Monthly
Notices Roy. Astronom. Soc. 85, 979-984 (1925).
THE INEQUALITY OF MILNE AND ITS CONVERSE
611
[2] G.H. Hardy, J.E. Littlewood and G. P61ya, Inequalities, Cambridge Univ. Press, Cambridge,
(1952).
[3] C.R. Rao, Statistical proofs of some matrix inequalities, Linear Algebra Appl. 321,307320 (2000).
[4] A. Kagan and C.R. Rao, Efficient Fisher score and its applications to matrices, Preprint,
(2000).
[5] R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge, (1985).
[6] R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge, (1991).