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volume 5, issue 3, article 73, 2004.

Received 14 February, 2004;

accepted 02 July, 2004.

Communicated by:P.S. Bullen

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Journal of Inequalities in Pure and Applied Mathematics

A NEW PROOF OF THE MONOTONICITY PROPERTY OF POWER MEANS

ALFRED WITKOWSKI

Mielczarskiego 4/29, 85-796 Bydgoszcz, Poland

EMail:[email protected]

2000c Victoria University ISSN (electronic): 1443-5756 109-04

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A New Proof of the Monotonicity Property of Power Means

Alfred Witkowski

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J. Ineq. Pure and Appl. Math. 5(3) Art. 73, 2004

IfM_ris the weighted power mean of the numbersx_j ∈[a, b]thenQ_r(a, b, x) = (a^r+b^r−M_r^r)^1/ris increasing inr. A new proof of this fact is given.

2000 Mathematics Subject Classification:26D15.

Key words: Convexity, Monotonicity, Power Means.

1. Introduction

Suppose that0 < a < b, a ≤ x₁ ≤ · · · ≤ x_n ≤ bandw_i are positive weights withP

wi = 1. The weighted power meansMr(x, w)of the numbersxi with weightsw_i are defined as

M_r(x, w) =X

w_ix^r_i¹_r

forr 6= 0, M₀(x, w) = expX

w_ilogx_i .

It is well-known (cf. [1,2,5]) thatMrincreases withrunless orxi are equal.

In [3] Mercer defined another family of functions

Q_r(a, b, x) = (a^r+b^r−M_r^r(x, w))^1/rforr6= 0, Q₀(a, b, x) =ab/M₀

and proved the following

Theorem 1.1. Forr < s Q_r(a, b, x)≤Q_s(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. Iff is convex then

(1.1) f

a+b−X w_ix_i

≤f(a) +f(b)−X

w_if(x_i).

For concavef the inequality reverses.

Our proof differs from the original one:

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Alfred Witkowski

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f(a+b−X

w_ix_i) =fX

w_i[(1−λ_i)a+λ_ib]

≤X

w_if([(1−λ_i)a+λ_ib])

≤X

wi[(1−λi)f(a) +λif(b)])

=X

w_i[f(a)−λ_if(a) +f(b)−(1−λ_i)f(b)]

=f(a) +f(b) +X

w_i[−λ_if(a)−(1−λ_i)f(b)]

≤f(a) +f(b)−X

wif(xi).

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Alfred Witkowski

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2. Proof of Theorem 1.1

Proof. Letea=a^r/Q^r_r, eb=b^r/Q^r_r, xe_i =x^r_i/Q^r_r. Applying (1.1) to the concave functionlogxwe obtain

0 = log

ea+eb−X w_ixe_i

≥logea+ logeb−X

w_ilogxe_i

=rlogQ₀ Qr

,

which shows that forr >0 Q−r ≤Q₀ ≤Q_r.

If0< r < sthen the functionf(x) = x^s/ris convex and from (1.1) we have

1 =f

ea+eb−X w_ixe_i

≤ a^s Q^s_r + b^s

Q^s_r −X w_ix^s_i

Q^s_r

= Q_s

Q_r s

,

soQ_r ≤Q_s.

Finally, forr < s <0 f is concave and we obtain1≥

Qs

Qr

s

also equivalent toQ_r ≤Q_s.

Obviously, equality holds if and only if allx_i’s are equalaor all are equalb.

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Alfred Witkowski

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[1] P.S. BULLEN, D.S. MITRINOVI ´CANDP.M. VASI ´C, Means and their In- equalities, D. Reidel, Dordrecht, 1998.

[2] G.H. HARDY, J.E. LITTLEWOOD ANDG. 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].

[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].