Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 5, Issue 3, Article 73, 2004
A NEW PROOF OF THE MONOTONICITY PROPERTY OF POWER MEANS
ALFRED WITKOWSKI
M IELCZARSKIEGO 4/29,
85-796 B YDGOSZCZ , P OLAND
alfred.witkowski@atosorigin.com
Received 14 February, 2004; accepted 02 July, 2004
Communicated by P.S. Bullen
A BSTRACT. If Mr is the weighted power mean of the numbers xj ∈ [a, b] then Qr (a, b, x) =
1/r
(ar + br − Mrr )
is increasing in r. A new proof of this fact is given.
Key words and phrases: Convexity, Monotonicity, Power Means.
2000 Mathematics Subject Classification. 26D15.
1. I NTRODUCTION
P
Suppose 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
r1
X
X
Mr (x, w) =
wi xri
for r 6= 0, M0 (x, w) = exp
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,
Q0 (a, b, x) = ab/M0
and proved the following
Theorem 1.1. For r < s
Qr (a, b, x) ≤ Qs (a, b, x).
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:
ISSN (electronic): 1443-5756
c 2004 Victoria University. All rights reserved.
109-04
2
A LFRED W ITKOWSKI
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 ).
2. P ROOF OF T HEOREM 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
a + eb −
wi xei ≤ s + s −
1=f e
wi is
Qr Qr
Qr
s
Qs
,
=
Qr
so Qr ≤ Qs .
s
Qs
Finally, for r < s < 0 f is concave and we obtain 1 ≥ Q
also equivalent to Qr ≤ Qs .
r
Obviously, equality holds if and only if all xi ’s are equal a or all are equal b.
R EFERENCES
[1] P.S. BULLEN, D.S. MITRINOVIĆ
Dordrecht, 1998.
AND
P.M. VASIĆ, Means and their Inequalities, D. Reidel,
[2] G.H. HARDY, J.E. LITTLEWOOD
Press, Cambridge, 1952.
AND
G. POLYA, Inequalities, 2nd ed. Cambridge University
[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].
[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].
[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].
J. Inequal. Pure and Appl. Math., 5(3) Art. 73, 2004
http://jipam.vu.edu.au/