Journal of Inequalities in Pure and Applied Mathematics

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1 Journal of Inequalities in Pure and Applied Mathematics STOLARSKY MEANS OF SEVERAL VARIABLES EDWARD NEUMAN Department of Mathematics Southern Illinois University Carbondale, IL , USA volume 6, issue 2, article 30, Received 19 October, 2004; accepted 24 February, Communicated by: Zs. Páles URL: Abstract Home Page c 2000 Victoria University ISSN (electronic):

2 Abstract A generalization of the Stolarsky means to the case of several variables is presented. 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 theorems for these means with the underlying measure being the Dirichlet measure are established Mathematics Subject Classification: 33C70, 26D20 Key words: Stolarsky means, Dresher means, Dirichlet averages, Totally positive functions, Inequalities The author is indebted to a referee for several constructive comments on the first draft of this paper. 1 Introduction and Notation Elementary Properties of E r,s (µ; X) The Mean E r,s (b; X) A Appendix: Total Positivity of Er,s r s (x, y) References Page 2 of 22

3 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 y) they are defined as follows [16] [ s r exp (1.1) E r,s (x, y) = [ x r y r ] 1 r s, rs(r s) 0; x r y r x s y s ( 1 r + xr ln x y r ln y x r y r ), r = s 0; ] 1 r, r 0, s = 0; r(ln x ln y) xy, r = s = 0. The mean E r,s (x, y) is symmetric in its parameters r and s and its variables x and y as well. Other properties of E r,s (x, y) include homogeneity of degree one in the variables x and y and monotonicity in r and s. It is known that E r,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]) (1.2) ln E r,s (x, y) = 1 s r s r ln I t dt Page 3 of 22

4 (r s), where I t I t (x, y) = E t,t (x, y) is the identric mean of order t. J. Pečarić and V. Šimić [15] have pointed out that [ 1 (1.3) E r,s (x, y) = 0 ] ( 1 tx s r s r s + (1 t)y s) s dt (s(r s) 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 will stand for the Euclidean simplex, which is defined by = { (u 1,..., u n 1 ) : u i 0, 1 i n 1, u u n 1 1 }. Further, let X = (x 1,..., x n ) be an n-tuple of positive numbers and let X min = min(x), X max = max(x). The following (1.4) L(X) = (n 1)! n i=1 x u i i du = (n 1)! exp(u Z) du Page 4 of 22

5 is the special case of the logarithmic mean of X which has been introduced in [9]. Here u = (u 1,..., u n 1, 1 u 1 u n 1 ) where (u 1,..., u n 1 ), du = du 1... du n 1, Z = ln(x) = (ln x 1,..., ln x n ), and x y = x 1 y x n y n is the dot product of two vectors x and y. Recently J. Merikowski [8] has proposed the following generalization of the Stolarsky mean E r,s to several variables (1.5) E r,s (X) = [ ] L(X r 1 ) r s L(X s ) (r s), where X r = (x r 1,..., x r n). In the paper cited above, the author did not prove that E r,s (X) is the mean of X, i.e., that (1.6) X min E r,s (X) X max holds true. If n = 2 and rs(r s) 0 or if r 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. The logarithmic mean L(µ; X) with the underlying measure µ is defined in [9] as follows (1.7) L(µ; X) = n i=1 x u i i µ(u) du = exp(u Z)µ(u) du. Page 5 of 22

6 We define [ L(µ; X r ) L(µ; X (1.8) E r,s (µ; X) = s ) exp ] 1 r s, [ d dr ln L(µ; Xr ) r s ], r = s. Let us note that for µ(u) = (n 1)!, the Lebesgue measure on, the first part of (1.8) simplifies to (1.5). In Section 2 we shall prove that E r,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 latter is denoted by µ b, where b = (b 1,..., b n ) R n +, and is defined as [2] (1.9) µ b (u) = 1 B(b) n i=1 u b i 1 i, where B( ) is the multivariate beta function, (u 1,..., u n 1 ), and u n = 1 u 1 u n 1. In the Appendix we shall prove that under certain conditions imposed on the parameters r and s, the function Er,s r s (x, y) is strictly totally positive as a function of x and y. Page 6 of 22

7 2. Elementary Properties of E r,s (µ; X) In order to prove that E r,s (µ; X) is a mean value we need the following version of the Mean-Value Theorem for integrals. Proposition 2.1. Let α := X min < X max =: β and let f, g C ( [α, β] ) with g(t) 0 for all t [α, β]. Then there exists ξ (α, β) such that E (2.1) n 1 f(u X)µ(u) du g(u X)µ(u) du = f(ξ) g(ξ). Proof. Let the numbers γ and δ and the function φ be defined in the following way γ = g(u X)µ(u) du, δ = f(u X)µ(u) du, φ(t) = γf(t) δg(t). Letting t = u X and, next, integrating both sides against the measure µ, we obtain φ(u X)µ(u) du = 0. On the other hand, application of the Mean-Value Theorem to the last integral gives φ(c X) µ(u) du = 0, Page 7 of 22

8 where c = (c 1,..., c n 1, c n ) with (c 1,..., c n 1 ) and c n = 1 c 1 c n 1. Letting ξ = c X and taking into account that µ(u) du = 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. For later use let us introduce the symbol E (p) r,s (µ; X) (p 0), where (2.2) E (p) r,s (µ; X) = [ E r,s (µ; X p ) ] 1 p. We are in a position to prove the following. Theorem 2.2. Let X R n + and let r, s R. Then (i) X min E r,s (µ; X) X max, (ii) E r,s (µ; λx) = λe r,s (µ; X), λ > 0, (λx := (λx 1,..., λx n )), (iii) E r,s (µ; X) increases with an increase in either r and s, (iv) ln E r,s (µ; X) = 1 r s r s ln E t,t(µ; X) dt, r s, (v) E (p) r,s (µ; X) = E pr,ps (µ; X), Page 8 of 22

9 (vi) E r,s (µ; X)E r, s (µ; X 1 ) = 1, (X 1 := (1/x 1,..., 1/x n )), (vii) E s r r,s (µ; X) = Er,p p r (µ; X)Ep,s s p (µ; X). Proof of (i). Assume first that r s. Making use of (1.8) and (1.7) we obtain E r,s (µ; X) = [ exp [ r(u Z) ] ] 1 r s µ(u) du exp [ s(u Z) ]. µ(u) du Application of (2.1) with f(t) = exp(rt) and g(t) = exp(st) gives E r,s (µ; X) = [ exp [ r(c Z) ] exp [ s(c Z) ] ] 1 r s = exp(c Z), where c = (c 1,..., c n 1, c n ) with (c 1,..., c n 1 ) and c n = 1 c 1 c n 1. Since c Z = c 1 ln x c n ln x n, ln X min c Z ln X max. This in turn implies that X min exp(c Z) X max. This completes the proof of (i) when r s. Assume now that r = s. It follows from (1.8) and (1.7) that ln E r,r (µ; X) = [ (u Z) exp [ r(u Z) ] µ(u) du exp [ r(u Z) ] µ(u) du Application of (2.1) to the right side with f(t) = t exp(rt) and g(t) = exp(rt) gives [ [ ]] (c Z) exp r(c Z) ln E r,r (µ; X) = exp [ r(c Z) ] = c Z. ]. Page 9 of 22

10 Since ln X min c Z ln X max, the assertion follows. This completes the proof of (i). Proof of (ii). The following result (2.3) L ( µ; (λx) r) = λ r L(µ; X r ) (λ > 0) is established in [9, (2.6)]. Assume that r s. Using (1.8) and (2.3) we obtain [ ] λ r L(µ; X r 1 ) r s E r,s (µ; λx) = = λer,s (µ; X). λ s L(µ; X s ) Consider now the case when r = s 0. Making use of (1.8) and (2.3) we obtain [ d E r,r (µ; λx) = exp dr ln L( µ; (λx) r) ] [ d = exp dr ln ( λ r L(µ; X r ) ) ] [ d ( = exp r ln λ + ln L(µ; X r ) )] = λe r,r (µ; X). dr When r = s = 0, an easy computation shows that (2.4) E 0,0 (µ; X) = n i=1 x w i i G(w; X), Page 10 of 22

11 where (2.5) w i = u i µ(u) du (1 i n) are called the natural weights or partial moments of the measure µ and w = (w 1,..., w n ). Since w w n = 1, E 0,0 (µ; λx) = λe 0,0 (µ; X). The proof of (ii) is complete. Proof of (iii). In order to establish the asserted property, let us note that the function r exp(rt) is logarithmically convex (log-convex) in r. This in conjunction with Theorem B.6 in [2], implies that a function r L(µ; X r ) is also log-convex in r. It follows from (1.8) that ln E r,s (µ; X) = ln L(µ; Xr ) ln L(µ; X s ) r s The right side is the divided difference of order one at r and s. Convexity of ln L(µ; X r ) in r implies that the divided difference increases with an increase in either r and s. This in turn implies that ln E r,s (µ; X) has the same property. Hence the monotonicity property of the mean E r,s in its parameters follows. Now let r = s. Then (1.8) yields ln E r,r (µ; X) = d dr [ ln L(µ; X r ) ]. Since ln L(µ; X r ) is convex in r, its derivative with respect to r increases with an increase in r. This completes the proof of (iii).. Page 11 of 22

12 Proof of (iv). Let r s. It follows from (1.8) that 1 r s r s r ln E t,t (µ; X) dt = 1 d [ ln L(µ; X t ) ] dt r s s dt = 1 [ ln L(µ; X r ) ln L(µ; X s ) ] r s = ln E r,s (µ; X). Proof of (v). Let r s. Using (2.2) and (1.8) we obtain E (p) r,s (µ; X) = [ E r,s (µ; X p ) ] 1 p = [ ] L(X pr 1 ) p(r s) L(X ps ) = Epr,ps (µ; X). Assume now that r = s 0. Making use of (2.2), (1.8) and (1.7) we have [ ] 1 E r,r (p) d (µ; X) = exp p dr ln L(µ; Xpr ) [ 1 = exp (u Z) exp [ pr(u Z) ] ] µ(u) du L(µ; X pr ) = E pr,pr (µ; X). The case when r = s = 0 is trivial because E 0,0 (µ; X) is the weighted geometric mean of X. Proof of (vi). Here we use (v) with p = 1 to obtain E r,s (µ; X 1 ) 1 = E r, s (µ; X). Letting X := X 1 we obtain the desired result. Page 12 of 22

13 Proof of (vii). There is nothing to prove when either p = r or p = s or r = s. In other cases we use (1.8) to obtain the asserted result. This completes the proof. In the next theorem we give some inequalities involving the means under discussion. Theorem 2.3. Let r, s R. Then the following inequalities (2.6) E r,r (µ; X) E r,s (µ; X) E s,s (µ; X) are valid provided r s. If s > 0, then (2.7) E r s,0 (µ; X) E r,s (µ; X). Inequality (2.7) is reversed if s < 0 and it becomes an equality if s = 0. Assume that r, s > 0 and let p q. Then (2.8) E (p) r,s (µ; X) E (q) r,s (µ; X) with the inequality reversed if r, s < 0. Proof. Inequalities (2.6) and (2.7) follow immediately from Part (iii) of Theorem 2.2. For the proof of (2.8), let r, s > 0 and let p q. Then pr qr and ps qs. Applying Parts (v) and (iii) of Theorem 2.2, we obtain E (p) r,s (µ; X) = E pr,ps (µ; X) E qr,qs (µ; X) = E (q) r,s (µ; X). When r, s < 0, the proof of (2.8) goes along the lines introduced above, hence it is omitted. The proof is complete. Page 13 of 22

14 3. The Mean E r,s (b; X) An important probability measure on is the Dirichlet measure µ b (u), b R n + (see (1.9)). Its role in the theory of special functions is well documented in Carlson s monograph [2]. When µ = µ b, the mean under discussion will be denoted by E r,s (b; X). The natural weights w i (see (2.5)) of µ b are given explicitly by (3.1) w i = b i /c (1 i n), where c = b b n (see [2, (5.6-2)]). For later use we define w = (w 1,..., w n ). Recall that the weighted Dresher mean D r,s (p; X) of order (r, s) R 2 of X R n + with weights p = (p 1,..., p n ) R n + is defined as [ n i=1 p ] ix r 1 r s i n, r s i=1 p ix s i (3.2) D r,s (p; X) = [ n i=1 exp p ] ix r i ln x i n i=1 p, r = s ix r i (see, e.g., [1, Sec. 24]). In this section we present two limit theorems for the mean E r,s with the underlying measure being the Dirichlet measure. In order to facilitate presentation we need a concept of the Dirichlet average of a function. Following [2, Def ] let Ω be a convex set in C and let Y = (y 1,..., y n ) Ω n, n 2. Further, let f be a measurable function on Ω. Define (3.3) F (b; Y ) = f(u Y )µ b (u) du. Page 14 of 22

15 Then F is called the Dirichlet average of f with variables Y = (y 1,..., y n ) and parameters b = (b 1,..., b n ). We need the following result [2, Ex ]. Let Ω be an open circular disk in C, and let f be holomorphic on Ω. Let Y Ω n, c C, c 0, 1,..., and w w n = 1. Then (3.4) lim c 0 F (cw; Y ) = n w i f(y i ), i=1 where cw = (cw 1,..., cw n ). We are in a position to prove the following. Theorem 3.1. Let w 1 > 0,..., w n > 0 with w w n = 1. If r, s R and X R n +, then lim c 0 + E r,s(cw; X) = D r,s (w; X). Proof. We use (1.7) and (3.3) to obtain L(cw; X) = F (cw; Z), where Z = ln X = (ln x 1,..., ln x n ). Making use of (3.4) with f(t) = exp(t) and Y = ln X we obtain Hence n lim L(cw; X) = c 0 + (3.5) lim c 0 + L(cw; Xr ) = i=1 w i x i. n w i x r i. Assume that r s. Application of (3.5) to (1.8) gives [ ] L(cw; X r 1 [ lim E ) r s n r,s(cw; X) = lim = i=1 w ix r i c 0 + c 0 + L(cw; X s ) n i=1 w ix s i i=1 ] 1 r s = Dr,s (w; X). Page 15 of 22

16 Let r = s. Application of (3.4) with f(t) = t exp(rt) gives n lim F (cw; Z) = c 0 + i=1 w i z i exp(rz i ) = n w i (ln x i )x r i. This in conjunction with (3.5) and (1.8) gives [ ] F (cw; Z) lim E r,r(cw; X) = lim exp c 0 + c 0 + L(cw; X r ) [ n i=1 = exp w ] ix r i ln x i n i=1 w ix r i This completes the proof. = D r,r (w; X). Theorem 3.2. Under the assumptions of Theorem 3.1 one has i=1 (3.6) lim c E r,s (cw; X) = G(w; X). Proof. The following limit (see [9, (4.10)]) (3.7) lim c L(cw; X) = G(w; X) will be used in the sequel. We shall establish first (3.6) when r s. It follows from (1.8) and (3.7) that [ ] L(cw; X r 1 lim E ) r s [ r,s(cw; X) = lim ] = G(w; X) r s 1 r s = G(w; X). c c L(cw; X s ) Page 16 of 22

17 Assume that r = s. We shall prove first that (3.8) lim c F (cw; Z) = [ ln G(w; X) ] G(w; X) r, where F is the Dirichlet average of f(t) = t exp(rt). Averaging both sides of we obtain f(t) = m=0 r m m! tm+1 (3.9) F (cw; Z) = m=0 r m m! R m+1(cw; Z), where R m+1 stands for the Dirichlet average of the power function t m+1. We will show that the series in (3.9) converges uniformly in 0 < c <. This in turn implies further that as c, we can proceed to the limit term by term. Making use of [2, )] we obtain R m+1 (cw; Z) Z m+1, m N, where Z = max { ln x i : 1 i n }. By the Weierstrass M test the series in (3.9) converges uniformly in the stated domain. Taking limits on both sides Page 17 of 22

18 of (3.9) we obtain with the aid of (3.4) lim F (cw; Z) = r m c m! lim R m+1(cw; Z) c m=0 ( n ) m+1 r m = w i z i m! m=0 i=1 = [ ln G(w; X) ] m=0 = [ ln G(w; X) ] m=0 = [ ln G(w; X) ] G(w; X) r. r m [ ] m ln G(w; X) m! 1 [ ] ln G(w; X) r m m! This completes the proof of (3.8). To complete the proof of (3.6) we use (1.8), (3.7), and (3.8) to obtain lim ln E F (cw; Z) r,r(µ; X) = lim c c L(cw; X r ) [ ] ln G(w; X) G(w; X) r = = ln G(w; X). G(w; X) r Hence the assertion follows. Page 18 of 22

19 A. Appendix: Total Positivity of E r s r,s (x, y) A real-valued function h(x, y) of two real variables is said to be strictly totally positive on its domain if every n n determinant with elements h(x i, y j ), where x 1 < x 2 < < x n and y 1 < y 2 < < y n is strictly positive for every n = 1, 2,... (see [5]). The goal of this section is to prove that the function Er,s r s (x, y) is strictly totally positive as a function of x and y provided the parameters r and s satisfy a certain condition. For later use we recall the definition of the R-hypergeometric function R α (β, β ; x, y) of two variables x, y > 0 with parameters β, β > 0 (A1) R α (β, β ; x, y) = Γ(β + β ) Γ(β)Γ(β ) (see [2, (5.9-1)]). 1 0 u β 1 (1 u) β 1 [ ux+(1 u)y ] α du Proposition A.1. Let x, y > 0 and let r, s R. If r < s, then E r s r,s (x, y) is strictly totally positive on R 2 +. Proof. Using (1.3) and (A1) we have (A2) E r s r,s (x, y) = R r s (1, 1; x s, y s ) (s(r s) 0). B. Carlson and J. Gustafson [3] have proven that R α (β, β ; x, y) is strictly totally positive in x and y provided β, β > 0 and 0 < α < β + β. Letting α = 1 r/s, β = β = 1, x := x s, y := y s, and next, using (A2) we obtain the desired result. s Page 19 of 22

20 Corollary A.2. Let 0 < x 1 < x 2, 0 < y 1 < y 2 and let the real numbers r and s satisfy the inequality r < s. If s > 0, then (A3) E r,s (x 1, y 1 )E r,s (x 2, y 2 ) < E r,s (x 1, y 2 )E r,s (x 2, y 1 ). Inequality (A3) is reversed if s < 0. Proof. Let a ij = E r s r,s (x i, y j ) (i, j = 1, 2). It follows from Proposition A.1 that det ( [a ij ] ) > 0 provided r < s. This in turn implies [ Er,s (x 1, y 1 )E r,s (x 2, y 2 ) ] r s > [ Er,s (x 1, y 2 )E r,s (x 2, y 1 ) ] r s. Assume that s > 0. Then the inequality r < s implies r s < 0. Hence (A3) follows when s > 0. The case when s < 0 is treated in a similar way. Page 20 of 22

21 References [1] E.F. BECKENBACH AND R. BELLMAN, Inequalities, Springer-Verlag, Berlin, [2] B.C. CARLSON, Special Functions of Applied Mathematics, Academic Press, New York, [3] B.C. CARLSON AND J.L. GUSTAFSON, Total positivity of mean values and hypergeometric functions, SIAM J. Math. Anal., 14(2) (1983), [4] P. CZINDER AND ZS. PÁLES An extension of the Hermite-Hadamard inequality and an application for Gini and Stolarsky means, J. Ineq. Pure Appl. Math., 5(2) (2004), Art. 42. [ONLINE: edu.au/article.php?sid=399]. [5] S. KARLIN, Total Positivity, Stanford Univ. Press, Stanford, CA, [6] E.B. LEACH AND M.C. SHOLANDER, Extended mean values, Amer. Math. Monthly, 85(2) (1978), [7] E.B. LEACH AND M.C. SHOLANDER, Multi-variable extended mean values, J. Math. Anal. Appl., 104 (1984), [8] J.K. MERIKOWSKI, Extending means of two variables to several variables, J. Ineq. Pure Appl. Math., 5(3) (2004), Art. 65. [ONLINE: http: //jipam.vu.edu.au/article.php?sid=411]. [9] E. NEUMAN, The weighted logarithmic mean, J. Math. Anal. Appl., 188 (1994), Page 21 of 22

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