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Volume 2010, Article ID 289730, 16 pagesdoi:10.1155/2010/289730 Research Article Differences of Weighted Mixed Symmetric Means and Related Results 1 Abdus Salam School of Mathematical Sc

Volume 2010, Article ID 289730, 16 pages

doi:10.1155/2010/289730

Research Article

Differences of Weighted Mixed Symmetric Means and Related Results

1 Abdus Salam School of Mathematical Sciences, GC University, 68-B, New Muslim Town,

Lahore 54600, Pakistan

2 Faculty of Textile Technology, University of Zagreb, Pierotti-jeva 6, 10000 Zagreb, Croatia

3 Faculty of Food Technology and Biotechnology, University of Zagreb, 10002 Zagreb, Croatia

Correspondence should be addressed to Khuram Ali Khan,khuramsms@gmail.com

Received 22 June 2010; Accepted 13 October 2010

Academic Editor: Marta Garc´ıa-Huidobro

Copyrightq 2010 Khuram Ali Khan 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

Some improvements of classical Jensen’s inequality are used to define the weighted mixed symmetric means Exponential convexity and mean value theorems are proved for the differences

of these improved inequalities Related Cauchy means are also defined, and their monotonicity is established as an application

1 Introduction and Preliminary Results

For n ∈ N, let x  x1, , x n  and p  p1, , p n  be positive n-tuples such thatn

i1p i  1



x1, , x n ; p1, , p n





⎧

⎪

⎪

⎪

⎪

i1

p i x r i

, r /  0,

i1x p i

i , r  0.

1.1

We introduce the mixed symmetric means with positive weights as follows:

M1s,t x, p; k 

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎛

⎝ 1

C n−1

k−1 1≤i 1< ···<i k ≤n

⎛

j1

p i j

⎞

t



x i1, x i k ; p i1, p i k⎞⎠1/s

, s /  0,



Π1≤i1< ···<i k ≤n

M t x i1, , x i k ; p i1, , p i kk j1p ij1/C n−1

k−1

, s  0.

1.2

We obtain the monotonicity of these means as a consequence of the following improvement

Theorem 1.1 Let I ⊆ R, x  x1, , x n  ∈ I n , p  p1, , p n  be a positive n-tuple such that

f k,n1 x, p : 1

C n−1

k−1 1≤i 1< ···<i k ≤n

⎛

j1

p i j

⎞

⎠f

⎛

⎝

j1p i j x i j

j1p i j

⎞

then

f k1 1,n x, p ≤ f1

that is

f

i1

p i x i

 f1

n,n x, p ≤ · · · ≤ f1

k,n x, p ≤ · · · ≤ f1

i1

If f is a concave function, then the inequality1.4 is reversed.

Corollary 1.2 Let s, t ∈ R such that s ≤ t, and let x and p be positive n-tuples such thatn

i1p i  1,

then, we have

M1t  M1

t,s x, p; 1 ≥ · · · ≥ M1

t,s x, p; k ≥ · · · ≥ M1

t,s x, p; n  M1

M1s  M1

s,t x, p; 1 ≤ · · · ≤ M1

s,t x, p; k ≤ · · · ≤ M1

s,t x, p; n  M1

Proof Let s, t ∈ R such that s ≤ t, if s, t / 0, then we set fx  x t/s , x i j  x s

i jin1.4 and raising

Let I ⊆ R be an interval, x, p be positive n-tuples such thatn

i1p i  1 Also let h, g :

M1h,g x, p; k  h−1

⎛

C n−1

k−1 1≤i 1< ···<i k ≤n

⎛

j1

p i j

⎞

⎛

⎜k j1p i j g

x i j

j1p i j

⎞

⎟

⎞

⎟

1.3

Corollary 1.3 By similar setting in 1.4, one gets the monotonicity of generalized means as follows:

h,g x, p; 1 ≥ · · · ≥ M1

h,g x, p; k ≥ · · · ≥ M1

where f  h ◦ g−1is convex and h is increasing, or f  h ◦ g−1is concave and h is decreasing;

M1

g,h x, p; 1 ≤ · · · ≤ M1

g,h x, p; k ≤ · · · ≤ M1

where f  g ◦ h−1is convex and g is decreasing, or f  g ◦ h−1is concave and g is increasing Remark 1.4 In fact Corollaries1.2and1.3are weighted versions of results in2

173 can be written in the following form:

Theorem 1.5 Let the conditions of Theorem 1.1 be satisfied for k ∈ N, 2 ≤ k ≤ n − 1, n ≥ 3 Then

f k,n1 x, p ≤ n − k

n− 1f 1,n1 x, p k− 1

where f1

Ω4

x, p; f n − k

n− 1f 1,n1 x, p k− 1

n− 1f n,n1 x, p − f1

Corollary 1.6 Let s, t ∈ R such that s ≤ t, and let x and p be positive n-tuples such thatn

i1p i  1.

Then, we have

M t t,s x, p; k ≤ n − k

n− 1M t t x, p k− 1

M s s,t x, p; k ≥ n − k

n− 1M s s x, p k− 1

Proof Let s, t ∈ R such that s ≤ t, if s, t / 0, then we set fx  x t/s , x i j  x s

i jin1.11 to obtain

1.13 and we set fx  x s/t , x i j  x t

i jin1.11 to obtain 1.14

Corollary 1.7 We set x i j  gx i j  and the convex function f  h ◦ g−1in1.11 to get

h

n− 1h



Theorem 1.8 Let f be a convex function defined on an interval I ⊆ R, x, p be positive n-tuples such

thatn

i1p i  1 and x1, , x n ∈ I Then

f

i1

p i x i

≤ · · · ≤ f2

k 1,n x, p ≤ f2

k,n x, p ≤ · · · ≤ f2

i1

where

C k n−1 k−11≤i1≤···≤i k ≤n

⎛

j1

p i j

⎞

⎠f

⎛

⎝

j1p i j x i j

j1p i j

⎞

If f is a concave function then the inequality1.16 is reversed.

follows:

M2s,t x, p; k 

⎧

⎪

⎪

⎨

⎪

⎪

⎩

⎛

⎝ 1

C n k k−1−1 1≤i1≤···≤i k ≤n

⎛

j1

p i j

⎞

t



x i1, , x i k ; p i1, p i k⎞⎠1/s

, s / 0;



Π1≤i1≤···≤i k ≤nM t x i1, , x i k ; p i1, p i kk j1p ij1/C n k−1

k−1

, s  0.

1.18

Corollary 1.9 Let s, t ∈ R such that s ≤ t, and let x and p be positive n-tuples such thatn

i1p i  1.

Then, we have

M2t  M2

t,s x, p; 1 ≥ · · · ≥ M2

t,s x, p; k ≥ · · · ≥ M2

M2s  M2

s,t x, p; 1 ≤ · · · ≤ M2

s,t x, p; k ≤ · · · ≤ M2

Proof Let s, t ∈ R such that s ≤ t, if s, t / 0, then we set fx  x t/s , x i j  x s

i j in1.16 and

i j in 1.16 and

We define the quasiarithmetic means with respect to1.17 as follows:

M2h,g x, p; k  h−1

⎛

C n k−1

k−1 1≤i 1≤···≤i k ≤n

⎛

j1

p i j

⎞

⎛

⎜k j1p i j g

x i j

j1p i j

⎞

⎟

⎞

⎟

to1.17

Corollary 1.10 By similar setting in 1.16, we get the monotonicity of these generalized means as

follows:

h,g x, p; 1 ≥ · · · ≥ M2

h,g x, p; k ≥ · · · ≥ M2

where f  h ◦ g−1is convex and h is increasing, or f  h ◦ g−1is concave and h is decreasing;

g,h x, p; 1 ≤ · · · ≤ M2

g,h x, p; k ≤ · · · ≤ M2

where f  g ◦ h−1is convex and g is decreasing, or f  g ◦ h−1is concave and g is increasing.

Theorem 1.11 Let M be a real linear space, U a non empty convex set in M, f : U → R a convex

function, and also let p be positive n-tuples such thatn

i1p i  1 and x1, , x n ∈ U Then

f

i1

p i x i

≤ · · · ≤ f3

k,n x, p ≤ · · · ≤ f3

where 1 ≤ k ≤ n and for I  {1, , n},

f3

k,n x, p 

i , ,i ∈I

p i1· · · p i k f

⎛

⎝ 1

k

k

j1

x i j

⎞

The mixed symmetric means with positive weights related to1.25 are

M3s,t x, p; k 

⎧

⎪

⎪

⎪

⎪

i1, ,i k ∈I

j1p i j M s t x i1, , x i k

, s /  0,

Πi1, ,i k ∈I M t x i1, , x i kΠk j1p ij, s  0.

1.26

Corollary 1.12 Let s, t ∈ R such that s ≤ t, and let x and p be positive n-tuples such thatn

i1p i  1.

Then, we have

M3t  M3

t,s x, p; 1 ≥ · · · ≥ M3

t,s x, p; k ≥ · · · ≥ M3

M3s  M3

s,t x, p; 1 ≤ · · · ≤ M3

s,t x, p; k ≤ · · · ≤ M3

Proof Let s, t ∈ R such that s ≤ t, if s, t / 0, then we set fx  x t/s , x i j  x s

i j in1.24 and

i j in 1.25 and

M3h,g x, p; k  h−1

⎛

⎝

i1, ,i k ∈I

p i1· · · p i k h ◦ g−1

⎛

⎝ 1

k

k

j1

g

x i j

⎞

⎠

⎞

to1.25

Corollary 1.13 By similar setting in 1.24, we get the monotonicity of generalized means as follows:

h,g x, p; 1 ≥ · · · ≥ M3

h,g x, p; k ≥ · · · ≥ M3

where f  h ◦ g−1is convex and h is increasing, or f  h ◦ g−1is concave and h is decreasing;

g,h x, p; 1 ≤ · · · ≤ M3

g,h x, p; k ≤ · · · ≤ M3

where f  g ◦ h−1is convex and g is decreasing, or f  g ◦ h−1is concave and g is increasing.

Theorem 1.14 Let I ⊆ R, f : I → R be a convex function, σ be an increasing function on 0, 1

such that1

f

0

≤

1

0

· · ·

1

0

f

1

k 1

k 1

i1

u x i

k 1

i1

dσ x i

≤

1

0

· · ·

1

0

f

1

k

k

i1

u x i

i1

dσ x i

≤ · · ·

≤

1

0

· · ·

1

0

f

1 2

2

i1

u x i



i1

dσ x i

≤

1

0

1.32

for all positive integers k.

1

0

· · ·

1

0

f

1

m

m

i1

u x i



i1

dσ x i −

1

0

· · ·

1

0

f

1

k

k

i1

u x i



i1

The mixed symmetric means with positive weights related to

1

0

· · ·

1

0

f

1

k

k

i1

u x i

i1

are defined as:

M5

s,t x; k 

⎧

⎪

⎪

⎪

⎪

0

· · ·

1

0

M s

t ux1, , ux kk

i1

dσ x i

, s /  0,

exp

0

· · ·

1

0

log M t ux1, , ux kk

i1

dσ x i

, s  0.

1.35

Corollary 1.15 Let s, t ∈ R such that s ≤ t, and let x and p be positive n-tuples such thatn

i1p i  1.

Then, we have

M5t  M5

t,s x, p; 1 ≥ · · · ≥ M5

t,s x, p; k ≥ · · · ≥ M5

M5s  M5

s,t x, p; 1 ≤ · · · ≤ M5

s,t x, p; k ≤ · · · ≤ M5

Proof Let s, t ∈ R such that s ≤ t, if s, t / 0, then we set fx  x t/s , u  u sin1.32 and raising

M5

h,g x; k  h−1 1

0

· · ·

1

0

h ◦ g−1

1

k

k

i1

g ◦ ux i



i1

dσ x i

h−1to1.34

Corollary 1.16 By similar setting in 1.32, we get the monotonicity of generalized means, given in

1.38:

h,g x, p; 1 ≥ · · · ≥ M5

h,g x, p; k ≥ · · · ≥ M5

where f  h ◦ g−1is convex and h is increasing, or f  h ◦ g−1is concave and h is decreasing;

M5

g,h x, p; 1 ≤ · · · ≤ M5

g,h x, p; k ≤ · · · ≤ M5

where f  g ◦ h−1is convex and g is decreasing, or f  g ◦ h−1is concave and g is increasing Remark 1.17 In fact unweighted version of these results were proved in6, but in Remark

For convex function f, we define

Ωi

x, p, f f i

m,n x, p − f i

1.41

Ωi

Ωi

for any convex function f.

The exponentially convex functions are defined in7 as follows.

Definition 1.18 A function f : a, b → R is exponentially convex if it is continuous and

n i,j1

ξ i ξ j f

x i x j



Proposition 1.19 Let f : a, b → R be a function, then following statements are equivalent;

i f is exponentially convex.

ii f is continuous and

n i,j1

ξ i ξ j f

i x j

2



for every ξ i ∈ R and every x i , x j ∈ a, b, 1 ≤ i, j ≤ n.

function.

⎧

⎪

⎪

⎪

⎪

x s

s s − 1 , s /  0, 1,

x log x, s  1.

1.46

φ s

⎧

⎪

⎪

1

s2e sx , s /  0,

1

2, s  0.

1.47

define the corresponding means of Cauchy type and establish their monotonicity

2 Main Result

Theorem 2.1 i Let the conditions of Theorem 1.1 be satisfied Consider

Ωi

t ϕ t

m,n−ϕ t

whereΩi

s is obtained by replacing convex function f with ϕ s for s ∈ R, in Ω i x, p, f i  1, , 5.

Then the following statements are valid.

s l s m /2p

l,m1is a positive semidefinite matrix Particularly

Ωi

s l s m /2

k

s is exponentially convex on R.

Proof. i Consider a function

l,m1

have

l,m1

u l u m x s lm−2,

l1

u l x s l /2−1

≥ 0.

2.4

l,m1

u l u m ϕ i s lm

m,n

−

l,m1

u l u m ϕ s lm

k,n

l,m1

u l u m

ϕ s lm

m,n−ϕ s lm

k,n

l,m1

u l u mΩi

s lm

2.5

s l s m /2p l,m1is a positive semidefinite, that is,2.2 is valid

s

Theorem 2.2. Theorem 2.1 is still valid for convex functions φ s  ϕ s

Ωi

s x, p; x2 / 0, then there exists ξ ∈ a, b such that

Ωi

x, p, f  1

Proof Since f ∈ C2a, b therefore there exist real numbers m  min x ∈a,b f x and M 

maxx ∈a,b f x It is easy to show that the functions φ1x, φ2x defined as

2

2.7

are convex

Ωi





≥ 0,

Ωi

x, p, x2 .

2.8

Ωi

x, p, fx − m2x2  0,

m

2Ωi

x, p, x2 ≤ Ωi

x, p, fx.

2.9

m

2Ωi

x, p, x2 ≤ Ωi

SinceΩi x, p, x2 / 0, therefore



x, p, fx

Hence, we have

Ωi

x, p, f 1

Theorem 2.4 Let n ≥ 3 and k be positive integer such that 2 ≤ k ≤ n − 1 and f, g ∈ C2a, b, then

there exists ξ ∈ a, b such that

Ωi

x, p, f

Ωi

x, p, g 

f ξ

provided that the denominators are non zero.

Proof Define h ∈ C2a, b in the way that

c1 Ωi

x, p, g

c2 Ωi

x, p, f.

2.15



c1

f ξ

g ξ

2



Ωi

k,n x, p, x2 / 0, therefore 2.16 gives

Ωi

x, p, f

Ωi

x, p, g 

f ξ

Corollary 2.5 Let x and p be positive n-tuples, then for distinct real numbers l and r, different from

zero and 1, there exists ξ ∈ a, b, such that

ξ l −r r r − 1

l l − 1

Ωi

x, p; x l

Proof Taking f x  x l and gx  x r, in 2.13, for distinct real numbers l and r, different

Remark 2.6 Since the function ξ → ξ l −r , l /  r is invertible, then from 2.18, we get

m≤

l l − 1

Ωi x, p; x l

Ωi x, p; x r

1/l−r

3 Cauchy Mean

ξ 

g

Ωi

x, p, f

Ωi

x, p, g

We have that the expression on the right hand side of above, is also a mean We define Cauchy means

M i l,r 

l l − 1

Ωi x, p; x l

Ωi x, p; x r

1/l−r

, r /  l, r, l / 0, 1,



Ωi x, p; ϕ l

Ωi x, p; ϕ r

1/l−r

, r /  l.

3.2

Also, we have continuous extensions of these means in other cases Therefore by limit, we have the following:

M i r,r  exp

Ωi

x, p; ϕ r ϕ0

Ωi

x, p; ϕ r



, r /  0, 1,

M 1,1 i  exp



x, p; ϕ o ϕ1 2Ωi

x, p; ϕ1



,

M i 0,0 exp



x, p; ϕ2 0

 2Ωi

x, p; ϕ0



.

3.3

Lemma 3.1 Let f be a convex function defined on an interval I ⊂ R and l ≤ v, r ≤ u, l / r, u / v.

Then

l − r ≤

inequality, as follows

r,l be given as in3.2 and r, l, u, v ∈ R such that r ≤ v, l ≤ u, then

M r,l i ≤ M i

Proof ByProposition 1.20Ωi

linLemma 3.1and get

l− log Ωi

r

l − r ≤

u

Corollary 3.3 Let x and p be positive n-tuples, then for distinct real numbers l, r, and s, all are

different from zero and 1, there exists ξ ∈ I, such that

ξ l −r  r r − s

l l − s

M i l,s x, p; k l−M i

l,s x, p; k 1 l



M i r,s x, p; kr−M i

Proof Set f x  x l/s and gx  x r/s , then taking x i → x s

Remark 3.4 Since the function ξ → ξ l −ris invertible, then from3.7 we get

m≤

⎛

l l − s

M i l,s x, p; k l−M l,s i x, p; k 1 l



M i r,s x, p; kr−M i

r,s x, p; k 1r

⎞

⎟

1/l−r

where l, r, and s are non zero, distinct real numbers.

The corresponding Cauchy means are given by

M i

l,r;s

⎛

l l − s

M i l,s x, p; k l−M i l,s x, p; k 1 l



M i r,s x, p; kr−M i

r,s x, p; k 1r

⎞

⎟

1/l−r

M l,r;s i 

Ωi

xs , p; ϕ l/s



Ωi

xs , p; ϕ r/s

 1/l−r

wherexs  x s

1, , x s

M i r,r;s exp

s ư 2r

Ωi

xs , p; ϕ r/s ϕ0



sΩi

xs , p; ϕ r/s



, r r ư s / 0, s / 0,

M i 0,0;s exp

1

s ư Ωi



xs , p; ϕ2 0



2sΩ i

xs , p; ϕ0



, s /  0,

M s,s;s i  exp

ư1

s ưΩi



xs , p; ϕ0ϕ1



2sΩ i

xs , p; ϕ1



, s /  0,

M i r,r;0 exp

ư2

r Ωi

 logx, p; xφ r



Ωi logx, p; φ r



, r /  0,

M i 0,0;0 exp

Ωi logx, p; xφ0

 3Ωi

logx, p; φ0



,

3.11

where logx  log x1, , log x n

Theorem 3.5 Let l, r, u, v ∈ R such that l ≤ v, r ≤ u, then

M i l,r;s ≤ M i

v,u;s , i  1, , n, 3.12

where M i

l,r is given in3.10.

Proof We takeΩi

byLemma 3.1for l, r, u, v ∈ R, l ≤ v, r ≤ u, we get

Ωi l

Ωi r

1/lưr

≤

Ωi v

Ωi u

1/vưu

For s > 0, we set x i  x s

by taking limit

Acknowledgments

This research was partially funded by Higher Education Commission, Pakistan The research

of the second author was supported by the Croatian Ministry of Science, Education and Sports under the Research Grant no 117-1170889-0888

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2010

⎞

The mixed symmetric means with positive weights related to1.25 are

M3s,t...

We introduce the mixed symmetric means with positive weights as follows: