Journal of Inequalities in Pure and Applied Mathematics

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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 MIELCZARSKIEGO4/29, 85-796 BYDGOSZCZ, POLAND

Received 14 February, 2004; accepted 02 July, 2004 Communicated by P.S. Bullen

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

Key words and phrases: Convexity, Monotonicity, Power Means.

2000 Mathematics Subject Classification. 26D15.

1. INTRODUCTION

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

w_i = 1.

The weighted power meansM_r(x, w)of the numbersx_iwith weightsw_i are defined as M_r(x, w) = X

w_ix^r_i¹_r

forr6= 0, M₀(x, w) = expX

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

In [3] Mercer defined another family of functions

Q_r(a, b, x) = (a^r+b^r−M_r^r(x, w))^1/rforr 6= 0, Q₀(a, b, x) = ab/M₀ and proved the following

Theorem 1.1. Forr < 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. 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:

ISSN (electronic): 1443-5756

109-04

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2 ALFREDWITKOWSKI

Proof. Letx_i =λ_ia+ (1−λ_ib). Then 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

w_i[(1−λ_i)f(a) +λ_if(b)])

=X

w_i[f(a)−λ_if(a) +f(b)−(1−λ_i)f(b)]

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

wi[−λif(a)−(1−λi)f(b)]

≤f(a) +f(b)−X

w_if(x_i).

2. PROOF OFTHEOREM1.1

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 functionlogx we obtain

0 = log

ea+eb−X w_ixe_i

≥logea+ logeb−X

w_ilogxe_i

=rlogQ₀ Q_r, which shows that forr >0 Q−r ≤Q₀ ≤Q_r.

If0< r < sthen the functionf(x) =x^s/r is 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

= Qs

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.

REFERENCES

[1] P.S. BULLEN, D.S. MITRINOVI ´C AND P.M. VASI ´C, 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].

[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/