Journal of Inequalities in Pure and
Applied Mathematics
A NEW PROOF OF THE MONOTONICITY PROPERTY
OF POWER MEANS
volume 5, issue 3, article 73,
2004.
ALFRED WITKOWSKI
Mielczarskiego 4/29,
85-796 Bydgoszcz, Poland
Received 14 February, 2004;
accepted 02 July, 2004.
Communicated by: P.S. Bullen
EMail: alfred.witkowski@atosorigin.com
Abstract
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Victoria University
ISSN (electronic): 1443-5756
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Abstract
If Mr is the weighted power mean of the numbers xj ∈ [a, b] then Qr (a, b, x) =
(ar + br − Mrr )1/r is increasing in r. A new proof of this fact is given.
2000 Mathematics Subject Classification: 26D15.
Key words: Convexity, Monotonicity, Power Means.
A New Proof of the Monotonicity
Property of Power Means
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
5
Alfred Witkowski
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1.
Introduction
Suppose
P that 0 < a < b , a ≤ x1 ≤ · · · ≤ xn ≤ b and wi are positive weights
with
wi = 1. The weighted power means Mr (x, w) of the numbers xi with
weights wi are defined as
Mr (x, w) =
X
wi xri
r1
for r 6= 0,
M0 (x, w) = exp
X
wi log xi .
It is well-known (cf. [1, 2, 5]) that Mr increases with r unless or xi are equal.
In [3] Mercer defined another family of functions
Qr (a, b, x) = (ar + br − Mrr (x, w))1/r for r 6= 0,
A New Proof of the Monotonicity
Property of Power Means
Alfred Witkowski
Q0 (a, b, x) = ab/M0
and proved the following
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Theorem 1.1. For r < s Qr (a, b, x) ≤ Qs (a, b, x).
Contents
The aim of this note is to give another proof of this theorem. We will use the
following version of the Jensen inequality ([4])
Lemma 1.2. If f is convex then
X
X
(1.1)
f a+b−
wi xi ≤ f (a) + f (b) −
wi f (xi ).
For concave f the inequality reverses.
Our proof differs from the original one:
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Proof. Let xi = λi a + (1 − λi b). Then
X
X
f (a + b −
wi xi ) = f
wi [(1 − λi )a + λi b]
X
≤
wi f ([(1 − λi )a + λi b])
X
≤
wi [(1 − λi )f (a) + λi f (b)])
X
=
wi [f (a) − λi f (a) + f (b) − (1 − λi )f (b)]
X
= f (a) + f (b) +
wi [−λi f (a) − (1 − λi )f (b)]
X
≤ f (a) + f (b) −
wi f (xi ).
A New Proof of the Monotonicity
Property of Power Means
Alfred Witkowski
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2.
Proof of Theorem 1.1
Proof. Let e
a = ar /Qrr , eb = br /Qrr , xei = xri /Qrr . Applying (1.1) to the concave
function log x we obtain
X
X
0 = log e
a + eb −
wi xei ≥ log e
a + log eb −
wi log xei
= r log
Q0
,
Qr
which shows that for r > 0 Q−r ≤ Q0 ≤ Qr .
If 0 < r < s then the function f (x) = xs/r is convex and from (1.1) we have
X
X xs
as
bs
e
1=f e
a+b−
wi xei ≤ s + s −
wi is
Q
Q
Qr
r s r
Qs
=
,
Qr
so Qr ≤ Qs .
s
Qs
Finally, for r < s < 0 f is concave and we obtain 1 ≥ Q
also equivalent
r
to Qr ≤ Qs .
Obviously, equality holds if and only if all xi ’s are equal a or all are equal b.
A New Proof of the Monotonicity
Property of Power Means
Alfred Witkowski
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References
[1] P.S. BULLEN, D.S. MITRINOVIĆ AND P.M. VASIĆ, Means and their Inequalities, D. Reidel, Dordrecht, 1998.
[2] G.H. HARDY, J.E. LITTLEWOOD AND G. POLYA, Inequalities, 2nd ed.
Cambridge University Press, Cambridge, 1952.
[3] A.McD. MERCER, A monotonicity property of power means, J. Ineq. Pure
and Appl. Math., 3(3) (2002), Article 40. [ONLINE: http://jipam.
vu.edu.au/article.php?sid=192].
A New Proof of the Monotonicity
Property of Power Means
[4] A.McD. MERCER, A variant of Jensen’s inequality, J. Ineq. Pure and Appl.
Math., 4(4) (2003), Article 73. [ONLINE: http://jipam.vu.edu.
au/article.php?sid=314].
Alfred Witkowski
[5] A. WITKOWSKI, A new proof of the monotonicity of power means, J.
Ineq. Pure and Appl. Math., 5(1) (2004), Article 6. [ONLINE: http://
jipam.vu.edu.au/article.php?sid=358].
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