Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 6, Issue 2, Article 30, 2005 STOLARSKY MEANS OF SEVERAL VARIABLES EDWARD NEUMAN DEPARTMENT OF MATHEMATICS SOUTHERN ILLINOIS UNIVERSITY CARBONDALE, IL 62901-4408, USA [email protected] URL: http://www.math.siu.edu/neuman/personal.html Received 19 October, 2004; accepted 24 February, 2005 Communicated by Zs. Páles ABSTRACT. A generalization of the Stolarsky means to the case of several variables is pre- sented. The new means are derived from the logarithmic mean of several variables studied in [9]. Basic properties and inequalities involving means under discussion are included. Limit the- orems for these means with the underlying measure being the Dirichlet measure are established. Key words and phrases: Stolarsky means, Dresher means, Dirichlet averages, Totally positive functions, Inequalities. 2000 Mathematics Subject Classification. 33C70, 26D20. 1. INTRODUCTION AND NOTATION In 1975 K.B. Stolarsky [16] introduced a two-parameter family of bivariate means named in mathematical literature as the Stolarsky means. Some authors call these means the extended means (see, e.g., [6, 7]) or the difference means (see [10]). For r, s ∈ R and two positive numbers x and y (x 6= y) they are defined as follows [16] 1 s xr − yr r−s , rs(r − s) 6= 0; s s r x − y 1 xr ln x − yr ln y exp − + , r = s 6= 0; r xr − yr (1.1) Er,s(x, y) = 1 xr − yr r , r 6= 0, s = 0; r(ln x − ln y) √ xy, r = s = 0. The mean Er,s(x, y) is symmetric in its parameters r and s and its variables x and y as well. Other properties of Er,s(x, y) include homogeneity of degree one in the variables x and y and ISSN (electronic): 1443-5756 c 2005 Victoria University. All rights reserved. The author is indebted to a referee for several constructive comments on the first draft of this paper. 201-04 2 EDWARD NEUMAN monotonicity in r and s. It is known that Er,s increases with an increase in either r or s (see [6]). It is worth mentioning that the Stolarsky mean admits the following integral representation ([16]) 1 Z s (1.2) ln Er,s(x, y) = ln It dt s − r r (r 6= s), where It ≡ It(x, y) = Et,t(x, y) is the identric mean of order t. J. Pecariˇ c´ and V. Šimic´ [15] have pointed out that 1 Z 1 r−s r−s s s s (1.3) Er,s(x, y) = tx + (1 − t)y dt 0 (s(r − s) 6= 0). This representation shows that the Stolarsky means belong to a two-parameter family of means studied earlier by M.D. Tobey [18]. A comparison theorem for the Stolarsky means have been obtained by E.B. Leach and M.C. Sholander in [7] and independently by Zs. Páles in [13]. Other results for the means (1.1) include inequalities, limit theorems and more (see, e.g., [17, 4, 6, 10, 12]). In the past several years researchers made an attempt to generalize Stolarsky means to several variables (see [6, 18, 15, 8]). Further generalizations include so-called functional Stolarsky means. For more details about the latter class of means the interested reader is referred to [14] and [11]. To facilitate presentation let us introduce more notation. In what follows, the symbol En−1 will stand for the Euclidean simplex, which is defined by En−1 = (u1, . , un−1): ui ≥ 0, 1 ≤ i ≤ n − 1, u1 + ··· + un−1 ≤ 1 . Further, let X = (x1, . , xn) be an n-tuple of positive numbers and let Xmin = min(X), Xmax = max(X). The following Z n Z Y ui (1.4) L(X) = (n − 1)! xi du = (n − 1)! exp(u · Z) du En−1 i=1 En−1 is the special case of the logarithmic mean of X which has been introduced in [9]. Here u = (u1, . , un−1, 1 − u1 − · · · − un−1) where (u1, . , un−1) ∈ En−1, du = du1 . dun−1, Z = ln(X) = (ln x1,..., ln xn), and x · y = x1y1 + ··· + xnyn is the dot product of two vectors x and y. Recently J. Merikowski [8] has proposed the following generalization of the Stolarsky mean Er,s to several variables 1 L(Xr) r−s (1.5) E (X) = r,s L(Xs) r r r (r 6= s), where X = (x1, . , xn). In the paper cited above, the author did not prove that Er,s(X) is the mean of X, i.e., that (1.6) Xmin ≤ Er,s(X) ≤ Xmax holds true. If n = 2 and rs(r − s) 6= 0 or if r 6= 0 and s = 0, then (1.5) simplifies to (1.1) in the stated cases. This paper deals with a two-parameter family of multivariate means whose prototype is given in (1.5). In order to define these means let us introduce more notation. By µ we will denote a probability measure on En−1. The logarithmic mean L(µ; X) with the underlying measure µ is J. Inequal. Pure and Appl. Math., 6(2) Art. 30, 2005 http://jipam.vu.edu.au/ STOLARSKY MEANS OF SEVERAL VARIABLES 3 defined in [9] as follows Z n Z Y ui (1.7) L(µ; X) = xi µ(u) du = exp(u · Z)µ(u) du. En−1 i=1 En−1 We define 1 L(µ; Xr) r−s , r 6= s L(µ; Xs) (1.8) Er,s(µ; X) = d exp ln L(µ; Xr) , r = s. dr Let us note that for µ(u) = (n − 1)!, the Lebesgue measure on En−1, the first part of (1.8) simplifies to (1.5). In Section 2 we shall prove that Er,s(µ; X) is the mean value of X, i.e., it satisfies inequalities (1.6). Some elementary properties of this mean are also derived. Section 3 deals with the limit theorems for the new mean, with the probability measure being the Dirichlet measure. The n latter is denoted by µb, where b = (b1, . , bn) ∈ R+, and is defined as [2] n 1 Y (1.9) µ (u) = ubi−1, b B(b) i i=1 where B(·) is the multivariate beta function, (u1, . , un−1) ∈ En−1, and un = 1 − u1 − · · · − un−1. In the Appendix we shall prove that under certain conditions imposed on the parameters r−s r and s, the function Er,s (x, y) is strictly totally positive as a function of x and y. 2. ELEMENTARY PROPERTIES OF Er,s(µ; X) In order to prove that Er,s(µ; X) is a mean value we need the following version of the Mean- Value Theorem for integrals. Proposition 2.1. Let α := Xmin < Xmax =: β and let f, g ∈ C [α, β] with g(t) 6= 0 for all t ∈ [α, β]. Then there exists ξ ∈ (α, β) such that R f(u · X)µ(u) du f(ξ) (2.1) En−1 = . R g(u · X)µ(u) du g(ξ) En−1 Proof. Let the numbers γ and δ and the function φ be defined in the following way Z Z γ = g(u · X)µ(u) du, δ = f(u · X)µ(u) du, En−1 En−1 φ(t) = γf(t) − δg(t). Letting t = u · X and, next, integrating both sides against the measure µ, we obtain Z φ(u · X)µ(u) du = 0. En−1 On the other hand, application of the Mean-Value Theorem to the last integral gives Z φ(c · X) µ(u) du = 0, En−1 J. Inequal. Pure and Appl. Math., 6(2) Art. 30, 2005 http://jipam.vu.edu.au/ 4 EDWARD NEUMAN where c = (c1, . , cn−1, cn) with (c1, . , cn−1) ∈ En−1 and cn = 1 − c1 − · · · − cn−1. Letting ξ = c · X and taking into account that Z µ(u) du = 1 En−1 we obtain φ(ξ) = 0. This in conjunction with the definition of φ gives the desired result (2.1). The proof is complete. The author is indebted to Professor Zsolt Páles for a useful suggestion regarding the proof of Proposition 2.1. (p) For later use let us introduce the symbol Er,s (µ; X) (p 6= 0), where 1 (p) p p (2.2) Er,s (µ; X) = Er,s(µ; X ) . We are in a position to prove the following. n Theorem 2.2. Let X ∈ R+ and let r, s ∈ R. Then (i) Xmin ≤ Er,s(µ; X) ≤ Xmax, (ii) Er,s(µ; λX) = λEr,s(µ; X), λ > 0, (λX := (λx1, . , λxn)), (iii) Er,s(µ; X) increases with an increase in either r and s, 1 (iv) ln E (µ; X) = R r ln E (µ; X) dt , r 6= s, r,s r − s s t,t (p) (v) Er,s (µ; X) = Epr,ps(µ; X), −1 −1 (vi) Er,s(µ; X)E−r,−s(µ; X ) = 1, (X := (1/x1,..., 1/xn)), s−r p−r s−p (vii) Er,s (µ; X) = Er,p (µ; X)Ep,s (µ; X). Proof of (i). Assume first that r 6= s. Making use of (1.8) and (1.7) we obtain 1 "R exp r(u · Z)µ(u) du# r−s E (µ; X) = En−1 . r,s R exp s(u · Z)µ(u) du En−1 Application of (2.1) with f(t) = exp(rt) and g(t) = exp(st) gives 1 " # r−s exp r(c · Z) E (µ; X) = = exp(c · Z), r,s exp s(c · Z) where c = (c1, .