Journal of Inequalities in Pure and Applied Mathematics STOLARSKY MEANS OF SEVERAL VARIABLES volume 6, issue 2, article 30, EDWARD NEUMAN 2005. Department of Mathematics Received 19 October, 2004; Southern Illinois University accepted 24 February, 2005. Carbondale, IL 62901-4408, USA Communicated by: Zs. Páles EMail: [email protected] URL: http://www.math.siu.edu/neuman/personal.html Abstract Contents JJ II J I Home Page Go Back Close c 2000 Victoria University ISSN (electronic): 1443-5756 Quit 201-04 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 vari- ables studied in [9]. Basic properties and inequalities involving means under discussion are included. Limit theorems for these means with the underlying measure being the Dirichlet measure are established. Stolarsky Means of Several Variables 2000 Mathematics Subject Classification: 33C70, 26D20 Key words: Stolarsky means, Dresher means, Dirichlet averages, Totally positive Edward Neuman functions, Inequalities The author is indebted to a referee for several constructive comments on the first draft of this paper. Title Page Contents Contents JJ II 1 Introduction and Notation ............................. 3 J I 2 Elementary Properties of Er,s(µ; X) ..................... 7 Go Back 3 The Mean Er,s(b; X) .................................. 14 r−s Close A Appendix: Total Positivity of Er,s (x, y) ................. 19 References Quit Page 2 of 22 J. Ineq. Pure and Appl. Math. 6(2) Art. 30, 2005 http://jipam.vu.edu.au 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 r r r−s s x − y , rs(r − s) 6= 0; Stolarsky Means of Several r xs − ys Variables r r 1 x ln x − y ln y Edward Neuman exp − + , r = s 6= 0; r xr − yr (1.1) Er,s(x, y) = 1 r r r Title Page x − y , r 6= 0, s = 0; r(ln x − ln y) Contents √ xy, r = s = 0. JJ II The mean Er,s(x, y) is symmetric in its parameters r and s and its variables x J I and y as well. Other properties of Er,s(x, y) include homogeneity of degree Go Back one in the variables x and y and monotonicity in r and s. It is known that E r,s Close 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]) Quit Page 3 of 22 1 Z s (1.2) ln E (x, y) = ln I dt r,s s − r t r J. Ineq. Pure and Appl. Math. 6(2) Art. 30, 2005 http://jipam.vu.edu.au (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 Stolarsky Means of Several and M.C. Sholander in [7] and independently by Zs. Páles in [13]. Other results Variables for the means (1.1) include inequalities, limit theorems and more (see, e.g., Edward Neuman [17, 4, 6, 10, 12]). In the past several years researchers made an attempt to generalize Stolarsky Title Page 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 Contents means the interested reader is referred to [14] and [11]. JJ II 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 J I Go Back En−1 = (u1, . , un−1): ui ≥ 0, 1 ≤ i ≤ n − 1, u1 + ··· + un−1 ≤ 1 . Close Further, let X = (x1, . , xn) be an n-tuple of positive numbers and let Xmin = Quit min(X), Xmax = max(X). The following Page 4 of 22 Z n Z Y ui (1.4) L(X) = (n − 1)! xi du = (n − 1)! exp(u · Z) du J. Ineq. Pure and Appl. Math. 6(2) Art. 30, 2005 En−1 i=1 En−1 http://jipam.vu.edu.au 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) Er,s(X) = s L(X ) Stolarsky Means of Several Variables (r 6= s), where Xr = (xr, . , xr ). In the paper cited above, the author did not 1 n Edward Neuman prove that Er,s(X) is the mean of X, i.e., that (1.6) Xmin ≤ Er,s(X) ≤ Xmax Title Page Contents 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. JJ II This paper deals with a two-parameter family of multivariate means whose J I prototype is given in (1.5). In order to define these means let us introduce more Go Back notation. By µ we will denote a probability measure on En−1. The logarithmic mean L(µ; X) with the underlying measure µ is defined in [9] as follows Close n Z Y Z Quit (1.7) L(µ; X) = xui µ(u) du = exp(u · Z)µ(u) du. i Page 5 of 22 En−1 i=1 En−1 J. Ineq. Pure and Appl. Math. 6(2) Art. 30, 2005 http://jipam.vu.edu.au 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). Stolarsky Means of Several Variables 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 Edward Neuman derived. Section 3 deals with the limit theorems for the new mean, with the probability measure being the Dirichlet measure. The latter is denoted by µb, Title Page n where b = (b1, . , bn) ∈ R+, and is defined as [2] Contents n 1 Y (1.9) µ (u) = ubi−1, JJ II b B(b) i i=1 J I where B(·) is the multivariate beta function, (u1, . , un−1) ∈ En−1, and un = Go Back 1−u −· · ·−u 1 n−1. In the Appendix we shall prove that under certain conditions Close r−s imposed on the parameters r and s, the function Er,s (x, y) is strictly totally positive as a function of x and y. Quit Page 6 of 22 J. Ineq. Pure and Appl. Math. 6(2) Art. 30, 2005 http://jipam.vu.edu.au 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) Stolarsky Means of Several En−1 Variables Proof. Let the numbers γ and δ and the function φ be defined in the following Edward Neuman way Z Z Title Page γ = g(u · X)µ(u) du, δ = f(u · X)µ(u) du, Contents En−1 En−1 φ(t) = γf(t) − δg(t). JJ II Letting t = u · X and, next, integrating both sides against the measure µ, we J I Go Back obtain Z φ(u · X)µ(u) du = 0. Close En−1 Quit On the other hand, application of the Mean-Value Theorem to the last integral gives Page 7 of 22 Z φ(c · X) µ(u) du = 0, E J. Ineq. Pure and Appl. Math. 6(2) Art. 30, 2005 n−1 http://jipam.vu.edu.au 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 re- Stolarsky Means of Several Variables garding the proof of Proposition 2.1.