Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 78175, 10 pages doi:10.1155/2007/78175 Research Article Schur-Convexity of Two Types of One-Parameter Mean Values in n Variables Ning-Guo Zheng, Zhi-Hua Zhang, and Xiao-Ming Zhang Received 10 July 2007; Revised 9 October 2007; Accepted 9 November 2007 Recommended by Simeon Reich We establish Schur-convexities of two types of one-parameter mean values in n variables. As applications, Schur-convexities of some well-known functions involving the complete elementary symmetric functions are obtained. Copyright © 2007 Ning-Guo Zheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Throughout the paper, R denotes the set of real numbers and R + denotes the set of strictly positive real numbers. Let n ≥ 2, n ∈ N, x = (x 1 ,x 2 , ,x n ) ∈ R n + ,andx 1/r = (x 1/r 1 ,x 1/r 2 , , x 1/r n ), where r ∈ R, r=0; let E n−1 ⊂ R n−1 be the simplex E n−1 =   u 1 , ,u n−1  : u i > 0(1 ≤ i ≤ n − 1), n−1  i=1 u i ≤ 1  , (1.1) and let dμ = du 1 , ,du n−1 be the differential of the volume in E n−1 . The weighted arithmetic mean A(x,u) and the power mean M r (x,u)oforderr with respect to the numbers x 1 ,x 2 , ,x n and the positive weights u 1 ,u 2 , ,u n with  n i =1 u i = 1 are defined, respectively, as A(x,u) =  n i =1 u i x i , M r (x,u) = (  n i =1 u i x r i ) 1/r for r=0, and M 0 (x,u)=  n i =1 x u i i . For u=(1/n,1/n, ,1/n), we denote A(x,u) Δ =A(x), M r (x,u) Δ =M r (x). The well-known logarithmic mean L(x 1 ,x 2 ) of two positive numbers x 1 and x 2 is L  x 1 ,x 2  = ⎧ ⎪ ⎨ ⎪ ⎩ x 1 − x 2 ln x 1 − ln x 2 , x 1 =x 2 , x 1 , x 1 = x 2 . (1.2) 2 Journal of Inequalities and Applications As further generalization of L(x 1 ,x 2 ), Stolarsky [1] studied the one-parameter mean, that is, L r  x 1 ,x 2  = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩  x r+1 1 − x r+1 2 (r +1)  x 1 − x 2   1/r , r=−1,0, x 1 =x 2 , x 1 − x 2 ln x 1 − ln x 2 , r =−1, x 1 =x 2 , 1 e  x x 1 1 x x 2 2  1/(x 1 −x 2 ) , r = 0, x 1 =x 2 , x 1 , x 1 = x 2 . (1.3) Alzer [2, 3] obtained another form of one-parameter mean, that is, F r  x 1 ,x 2  = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ r r +1 · x r+1 1 − x r+1 2 x r 1 − x r 2 , r=−1,0, x 1 =x 2 , x 1 x 2 · ln x 1 − ln x 2 x 1 − x 2 , r =−1, x 1 =x 2 , x 1 − x 2 ln x 1 − ln x 2 , r = 0, x 1 =x 2 , x 1 , x 1 = x 2 . (1.4) These two means can be written also as L r  x 1 ,x 2  = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩   1 0  x 1 u +x 2 (1 − u)  r du  1/r , r=0, exp   1 0 ln  x 1 u +x 2 (1 − u)  du  , r = 0, F r  x 1 ,x 2  = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩  1 0  x r 1 u +x r 2 (1 − u)  1/r du, r=0,  1 0 x u 1 x 1−u 2 du, r = 0. (1.5) Correspondingly, Pittenger [4] and Pearce et al. [5] investigated the means above in n variables, respectively, L r (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩  (n − 1)!  E n−1  A(x,u)  r dμ  1/r , r=0, exp  (n − 1)!  E n−1 ln A(x,u)dμ  , r = 0, F r (x) = (n − 1)!  E n−1 M r (x,u)dμ, (1.6) where u n = 1 −  n−1 i =1 u i . Ning-Guo Zheng et al. 3 Expressions (1.3)and(1.4) can be also written by using 2-order determinants, that is, L r  x 1 ,x 2  = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎝ 1 r +1 ·       1 x r+1 2 1 x r+1 1              1 x 2 1 x 1       ⎞ ⎠ 1/r , r=−1,0, x 1 =x 2 ,       1 x 2 1 x 1              1lnx 2 1lnx 1       , r =−1, x 1 =x 2 , exp ⎧ ⎨ ⎩ ⎛ ⎝       1 x 2 ln x 2 1 x 1 ln x 1              1 x 2 1 x 1       ⎞ ⎠ − 1 ⎫ ⎬ ⎭ , r = 0, x 1 =x 2 , x 1 , x 1 = x 2 , F r  x 1 ,x 2  = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎝ r       1 x r+1 2 1 x r+1 1       ⎞ ⎠  ⎛ ⎝ (r +1)       1 x r 2 1 x r 1       ⎞ ⎠ , r=−1,0, x 1 =x 2 , x 1 x 2       1lnx 2 1lnx 1              1 x 2 1 x 1       , r =−1, x 1 =x 2 ,       1 x 2 1 x 1              1lnx 2 1lnx 1       , r = 0, x 1 =x 2 , x 1 , x 1 = x 2 . (1.7) Utilizing higher-order generalized Vandermonde determinants, Xiao et al. [8, 7, 6, 9]gave the analogous definitions of L r (x)andF r (x). Obviously, L r (x)andF r (x) are symmetric with respect to x 1 ,x 2 , ,x n , r → L r (x)and r → F r (x) are continuous for any x ∈ R n + . In [4, 5, 10, 11], the authors studied the Schur-convexities of L r (x 1 ,x 2 )andF r (x 1 ,x 2 ). In this paper, we establish the Schur-convexities of two types of one-parameter mean values L r (x)andF r (x) for several positive numbers. As applications, Schur-convexities of some well-known functions involving the complete elementary symmetric functions are obtained. 2. Some definitions and lemmas The Schur-convex function was introduced by Schur [12] in 1923, and has many impor- tant applications in analytic inequalities. The following definitions can be found in many references such as [12–17]. Definit ion 2.1. For u = (u 1 ,u 2 , ,u n ), v = (v 1 ,v 2 , ,v n ) ∈ R n , without loss of general- ity, it is assumed that u 1 ≥ u 2 ≥ ··· ≥ u n and v 1 ≥ v 2 ≥ ··· ≥ v n .Thenu is said to be majorized by v (in symbols u ≺ v)if  k i =1 u i ≤  k i =1 v i for k = 1,2, ,n − 1and  n i =1 u i =  n i =1 v i . Definit ion 2.2. Let Ω ⊂ R n . A function ϕ : Ω → R is said to be a Schur-convex (Schur- concave) function if u ≺ v implies ϕ(u) ≤ (≥ )ϕ(v). 4 Journal of Inequalities and Applications Every Schur-convex function is a symmetric function [18]. But it is not hard to see that not every symmetric function can be a Schur-convex function [15, page 258]. However, we have the follow ing so-called Schur condition. Lemma 2.3 [12, page 57]. Suppose that Ω ⊂ R n is symmetric w ith respect to permutations and convexset, and has a nonempty interior set Ω 0 .Letϕ : Ω → R be continuous on Ω and continuously differentiable in Ω 0 .Then,ϕ is a Schur-convex (Schur-concave) function if and only if it is symmetric and if  u 1 − u 2   ∂ϕ ∂u 1 − ∂ϕ ∂u 2  ≥ (≤ )0 (2.1) holds for any u = (u 1 ,u 2 , ,u n ) ∈ Ω 0 . Lemma 2.4. Let m ≥ 1,n ≥ 2, m, n ∈ N, Λ ⊂ R m , Ω ⊂ R n , φ : Λ × Ω → R, φ(v,x) be con- tinuous with respect to v ∈ Λ for any x ∈ Ω.LetΔ be a set of all v ∈ Λ such that the function x → φ(v,x) is a Schur-convex (Schur-concave) function. Then Δ is a closed s et of Λ. Proof. Let l ≥ 1, l ∈ N, v l ∈ Δ, v 0 ∈ Λ, v l →v 0 if l→ + ∞.AccordingtoDefinition 2.2, φ(v l ,y) ≥ (≤ )φ(v l ,z)holdsforanyy,z ∈ Ω and y  z.Letl→ + ∞,thenwehaveφ(v 0 ,y) ≥ (≤ )φ(v 0 ,z). Hence v 0 ∈ Δ,soΔ is a closed set of Λ.  3. Main results Theorem 3.1. Given r ∈ R, L r (x) is Schur-convex if r ≥ 1 and Schur-concave if r ≤ 1. Proof. Denote u = (u 2 ,u 1 ,u 3 , ,u n ). If r =0, owing to the symmetry of L r (x) with respect to x 1 ,x 2 , ,x n ,wehave g r (x)   E n−1  A(x,u)  r dμ =  E n−1  A  x, u  r dμ. (3.1) Therefore, ∂g r ∂x 1 = r  E n−1 u 1  A(x,u)  r−1 dμ = r  E n−1 u 2  A  x, u  r−1 dμ, ∂g r ∂x 2 = r  E n−1 u 1  A  x, u  r−1 dμ = r  E n−1 u 2  A(x,u)  r−1 dμ. (3.2) It follows that ∂g r ∂x 1 − ∂g r ∂x 2 = r  E n−1 u 1   A(x,u)  r−1 −  A  x, u  r−1  dμ, ∂g r ∂x 1 − ∂g r ∂x 2 = r  E n−1 u 2   A  x, u  r−1 −  A(x,u)  r−1  dμ. (3.3) By combining (3.3) with (3.2), we have ∂g r ∂x 1 − ∂g r ∂x 2 = r 2  E n−1  u 1 − u 2    A(x,u)  r−1 −  A  x, u  r−1  dμ. (3.4) Ning-Guo Zheng et al. 5 By Lagrange’s mean value theorem, we find that  A(x,u)  r−1 −  A  x, u  r−1 = (r − 1)  x 1 u 1 + x 2 u 2 − x 2 u 1 − x 1 u 2   ξ + n  i=3 u i x i  r−2 = (r − 1)  u 1 − u 2  x 1 − x 2   ξ + n  i=3 u i x i  r−2 , (3.5) where ξ is between x 1 u 1 + x 2 u 2 and x 2 u 1 + x 1 u 2 . From (3.4)and(3.5), we have  x 1 − x 2   ∂g r ∂x 1 − ∂g r ∂x 2  = r(r − 1) 2  x 1 − x 2  2 S r (x), (3.6) where S r (x) =  E n−1  u 1 − u 2  2  ξ + n  i=3 u i x i  r−2 dμ ≥ 0. (3.7) Hence, for r =0, we get  x 1 − x 2   ∂L r ∂x 1 − ∂L r ∂x 2  = (n − 1)!· 1 r ·  L r  1−r ·  x 1 − x 2   ∂g r ∂x 1 − ∂g r ∂x 2  = (n − 1)!· r − 1 2 ·  L r  1−r ·  x 1 − x 2  2 S r (x). (3.8) From Lemma 2.3, it is clear that L r is Schur-convex for r>1 and Schur-concave for r<1 and r =0. According to Lemma 2.4 and the continuity of r → L r (x), let r→0,1−,or1+inL r (x), we know that L 0 (x) is a Schur-concave function, and L 1 (x) is both a Schur-concave func- tion and a Schur-convex function.  Theorem 3.2. Given r ∈ R, F r (x) is Schur-convex if r ≥ 1 and Schur-concave if r ≤ 1. Proof. Denote u = (u 2 ,u 1 ,u 3 , ,u n ). For r=0, F r (x) = (n − 1)!  E n−1 M r (x,u)dμ = (n − 1)!  E n−1 M r  x, u  dμ, (3.9) ∂F r ∂x 1 = (n − 1)!  E n−1 x r−1 1 u 1  M r (x, u )  1−r dμ = (n − 1)!  E n−1 u 1  M r (x, u ) x 1  1−r dμ, (3.10) ∂F r ∂x 2 = (n − 1)!  E n−1 x r−1 2 u 1  M r  x, u  1−r dμ = (n − 1)!  E n−1 u 1  M r  x, u  x 2  1−r dμ. (3.11) 6 Journal of Inequalities and Applications Combination of (3.10)with(3.11)yields ∂F r ∂x 1 − ∂F r ∂x 2 = (n − 1)!  E n−1 u 1  M r (x,u) x 1  1−r −  M r  x, u  x 2  1−r  dμ. (3.12) By using the mean value theorem, we find  M r (x,u) x 1  1−r −  M r  x, u  x 2  1−r =  u 1 + u 2 x r 2 +  n i =3 u i x r i x r 1  (1−r)/r −  u 1 + u 2 x r 1 +  n i =3 u i x r i x r 2  (1−r)/r = 1 − r r  u 2 x r 2 +  n i =3 u i x r i x r 1 − u 2 x r 1 +  n i =3 u i x r i x r 2   u 1 + θ 1  (1−2r)/r = 1 − r r · u 2 x 2r 2 + x r 2  n i =3 u i x r i − u 2 x 2r 1 − x r 1  n i =3 u i x r i x r 1 x r 2 ·  u 1 + θ 1  (1−2r)/r = (1 − r)  x 2 − x 1  u 1 + θ 1  (1−2r)/r T  x,u;θ 2  , (3.13) where θ 1 is between (u 2 x r 2 +  n i =3 u i x r i )/x r 1 and (u 2 x r 1 +  n i =3 u i x r i )/x r 2 , θ 2 is between x 1 and x 2 ,andT(x,u;θ 2 ) = (2u 2 θ 2r−1 2 + θ r−1 2  n i =3 u i x r i )/x r 1 x r 2 ≥ 0. From (3.12)and(3.13), we have  x 1 − x 2   ∂F r ∂x 1 − ∂F r ∂x 2  = (r − 1)  x 1 − x 2  2 (n − 1)!  E n−1 u 1  u 1 + θ 1  (1−2r)/r T  x,u;θ 2  dμ. (3.14) It follows that F r is Schur-convex for r>1 and Schur-concave for r<1andr=0by Lemma 2.3. According to Lemma 2.4 and the continuity of r → F r (x), let r→0,1−,or1+inF r (x). We know that F 0 (x) is a Schur-concave function, and F 1 (x) is both a Schur-concave func- tion and a Schur-convex function.  Theorem 3.3. L r (x 1/r ) and F r (x 1/r ) are Schur-concave functions if r ≥ 1,andSchur-convex functions if r ≤ 1 and r=0. Ning-Guo Zheng et al. 7 Proof. We can easily obtain that L r  x 1/r  =  (n − 1)!  E n−1 M 1/r (x,u)dμ  1/r = F 1/r 1/r (x), F r  x 1/r  = (n − 1)!  E n−1  A(x,u)  1/r dμ = L r 1/r (x), (3.15)  x 1 − x 2   ∂L r  x 1/r  ∂x 1 − ∂L r  x 1/r  ∂x 2  = 1 r  x 1 − x 2   ∂F 1/r (x) ∂x 1 − ∂F 1/r (x) ∂x 2  · F (1−r)/r 1/r (x),  x 1 − x 2   ∂F r  x 1/r  ∂x 1 − ∂F r  x 1/r  ∂x 2  = r  x 1 − x 2   ∂L 1/r (x) ∂x 1 − ∂L 1/r (x) ∂x 2  · L r−1 1/r (x). (3.16) From Theorems 3.1 and 3.2, we know that both L 1/r (x)andF 1/r (x)areSchur-concave functions if r ≥1 and Schur-convex functions if 0 <r≤1orr<0. According to Lemma 2.3 and (3.16), the required result of Theorem 3.3 is proved.  4. Applications As applications of the theorems above, we have the following corollaries. Corollary 4.1 (See [19, Theorem 3.1] and [12, page 82]). For r ≥ 1, r ∈ N,thecomplete elementary symmetric function C r (x) =  i 1 +i 2 +···+i n =r, i 1 , ,i n ≥0 are integers x i 1 1 x i 2 2 , ,x i n n (4.1) is Schur-convex. Proof. If r ≥ 1, r ∈ N, then (see [20, page 164]) C r (x) =  n − 1+r r  L r r (x). (4.2) By Theorem 3.1 and Lemma 2.3, it is easy to see that L r r (x) is a Schur-convex function. Therefore, C r (x) is a Schur-convex function.  Corollary 4.2. The complete symmetric function of the first degree: D r (x) =  i 1 +i 2 +···+i n =r, i 1 , ,i n ≥0 are integers  x i 1 1 x i 2 2 , ,x i n n  1/r (4.3) (see [6,Theorem5]and[9]), is Schur-concave for r ≥ 1, r ∈ N. 8 Journal of Inequalities and Applications Proof. If r ≥ 1, r ∈ N, then we have (see [6,Theorem5]) D r (x) =  n − 1+r r  F 1/r (x). (4.4) By considering Theorem 3.2, we prove the required result.  Corollary 4.3. Let r=0, x,y ∈ R n + , x r  y r . Then L r (x) ≤ L r (y) and F r (x) ≤ F r (y) if r ≥ 1. They are reversed if r ≤ 1 and r=0. Proof. Suppose r ≥ 1(r ≤ 1,r=0). L r (x 1/r ) is a Schur-concave (Schur-convex) function by Theorem 3.3.Then L r   x r  1/r  ≤ (≥ )L r   y r  1/r  , L r (x) ≤ (≥ )L r (y). (4.5) For F r (x 1/r ), the proof is similar; we omit the details.  Corollary 4.4. If r ≥ 1, then A(x) ≤ L r (x) ≤ M r (x), A(x) ≤ F r (x) ≤ M r (x). (4.6) Inequalities (4.6) are reversed if r ≤ 1. Proof. If r ≥ 1, owing to Theorem 3.1 and  x 1 ,x 2 , ,x n    A(x),A(x), ,A(x)   A(x), (4.7) we have L r (x) ≥ L r  A(x)  =  (n − 1)!  E n−1  n  i=1 A(x)u i  r dμ  1/r = A(x)  (n − 1)!  E n−1  n  i=1 u i  r dμ  1/r = A(x). (4.8) Obviously, if r ≤ 1, r=0, inequality (4.8)isreversedbyTheorem 3.1.Forr = 0, because of the continuity of r → L r (x), we have L 0 (x) ≤ A(x). By the same way, we find that F r (x) ≥ A(x)ifr ≥ 1, and F r (x) ≤ A(x)ifr ≤ 1. In addi- tion, x r =  x r 1 ,x r 2 , ,x r n    M r r (x),M r r (x), ,M r r (x)    M r (x),M r (x), ,M r (x)  r   M r (x)  r . (4.9) Ning-Guo Zheng et al. 9 If r ≥ 1, according to Corollary 4.3,weget L r (x) ≤ L r  M r (x)  =  (n − 1)!  E n−1  n  i=1 M r (x)u i  r dμ  1/r = M r (x). (4.10) If r ≤ 1, inequality (4.10)isobviouslyreversedbyCorollary 4.3 again. Similarly, we have F r (x) ≤ M r (x)ifr ≥ 1, and F r (x) ≥ M r (x)ifr ≤ 1.  Acknowledgments This work was supported by the NSF of Zhejiang Broadcast and TV University under Grant no. XKT-07G19. The authors are grateful to the referees for their valuable sugges- tions. References [1] K. B. Stolarsky, “Generalizations of the logarithmic mean,” Mathematics Magazine, vol. 48, pp. 87–92, 1975. [2] H. Alzer, “ ¨ Uber eine einparametrige familie von mittelwerten [On a one-parameter family of means],” Bayerische Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse. Sitzungsberichte, pp. 1–9, 1987 (German). [3] H. Alzer, “ ¨ Uber eine einparametrige Familie von Mittelwerten. II. [On a one-parameter family of mean values. II],” Bayerische Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse. Sitzungsberichte, pp. 23–39, 1988 (German). [4] A. O. Pittenger, “The logarithmic mean in n variables,” The American Mathematical Monthly, vol. 92, no. 2, pp. 99–104, 1985. [5] C.E.M.Pearce,J.Pe ˇ cari ´ c, and V. ˇ Simi ´ c, “On weighted generalized logarithmic means,” Houston Journal of Mathematics, vol. 24, no. 3, pp. 459–465, 1998. [6] Z G. Xiao and Z H. Zhang, “The Heron mean of n positive numbers,” Journal of YueYang Nor- mal University, vol. 14, no. 2, pp. 1–5, 2001 (Chinese). [7] Z G. Xiao, Z H. Zhang, and F. Qi, “A type of mean values of several positive numbers with two parameters,” RGMIA Research Report Collection, vol. 9, no. 2, article 11, 2006. [8] Z G. Xiao and Z H. Zhang, “The Stolarsky mean of n positive numbers,” Journal of YueYang Norm al University, vol. 14, no. 4, pp. 5–8, 2001 (Chinese). [9] Z G. Xiao, Z H. Zhang, and V. Lokesha, “The weighted Heron mean of several positive num- bers,” RGMIA Research Report Collection, vol. 8, no. 3, article 6, 2005. [10] N. Elezovi ´ candJ.Pe ˇ cari ´ c, “A note on Schur-convex functions,” The Rocky Mountain Journal of Mathematics, vol. 30, no. 3, pp. 853–856, 2000. [11] F. Qi, “A note on Schur-convexity of extended mean values,” The Rocky Mountain Journal of Mathematics, vol. 35, no. 5, pp. 1787–1793, 2005. [12] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorizat ion and Its Applications, vol. 143 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1979. [13] J C. Kuang, Applied Inequalities, Shandong Science and Technology Press, Jinan, China, 3rd edi- tion, 2004. [14] D. S. Mitrinovi ´ c, Analytic Inequalities, vol. 1965 of Die Grundlehren der mathematischen Wisen- schaften, Springer, New York, NY, USA, 1970. 10 Journal of Inequalities and Applications [15] A. W. Roberts and D. E. Varberg, Convex Functions. Pure and Applied Mathematics, vol. 57, Aca- demic Press, New York, NY, USA, 1973. [16] B Y. Wang, Foundations of Majorization Inequalities, Beijing Normal University Press, Beijing, China, 1990. [17] X M. Zhang , Geometrically Convex Functions, An’hui University Press, Hefei, China, 2004. [18] X M. Zhang, “Optimization of Schur-convex functions,” Mathematical Inequalities & Applica- tions, vol. 1, no. 3, pp. 319–330, 1998. [19] K. Guan, “Schur-convexity of the complete elementary symmetric function,” Journal of Inequal- ities and Applications, vol. 2006, Article ID 67624, 9 pages, 2006. [20] G. H. Hardy, J. E. Littlewood, and G. P ´ olya, Inequalities, Cambridge University Press, Cam- bridge, UK, 2nd edition, 1952. Ning-Guo Zheng: Huzhou Broadcast and TV University, Huzhou, Zhejiang 313000, China Email address: yczng@hzvtc.net Zhi-Hua Zhang: Zixing Educational Research Section, Chenzhou, Hunan 423400, China Email address: zxzh1234@163.com Xiao-Ming Zhang: Haining College, Zhejiang Broadcast and TV University, Haining, Zhejiang 314400, China Email address: zjzxm79@tom.com . Schur-convexities of two types of one-parameter mean values in n variables. As applications, Schur-convexities of some well-known functions involving the complete elementary symmetric functions are. elementary symmetric functions are obtained. 2. Some definitions and lemmas The Schur-convex function was introduced by Schur [12] in 1923, and has many impor- tant applications in analytic inequalities the Schur-convexities of two types of one-parameter mean values L r (x)andF r (x) for several positive numbers. As applications, Schur-convexities of some well-known functions involving the complete