Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID pptx

Pe ˇcari ´c2, 4 1 Department of Mathematics, University of Pannonia, University Street 10, 8200 Veszpr´em, Hungary 2 Abdus Salam School of Mathematical Sciences, GC University, 68-B, New

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Volume 2011, Article ID 350973, 19 pages

doi:10.1155/2011/350973

Research Article

Refinements of Results about Weighted Mixed

Symmetric Means and Related Cauchy Means

L ászl ó Horv áth,1 Khuram Ali Khan,2, 3 and J Pe ˇcari ć2, 4

1 Department of Mathematics, University of Pannonia, University Street 10,

8200 Veszpr´em, Hungary

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

Lahore 54600, Pakistan

3 Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan

4 Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia

Received 26 November 2010; Accepted 23 February 2011

Academic Editor: Michel Chipot

Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

A recent refinement of the classical discrete Jensen inequality is given by Horv´ath and Peˇcari´c

In this paper, the corresponding weighted mixed symmetric means and Cauchy-type means are defined We investigate the exponential convexity of some functions, study mean value theorems, and prove the monotonicity of the introduced means

1 Introduction and Preliminary Results

A new refinement of the discrete Jensen inequality is given in1 The following notations are also introduced in1

Let X be a set, P X its power set, and |X| denotes the number of elements in X Let

u ≥ 1 and v ≥ 2 be fixed integers Define the functions

S v,w:{1, , u} v −→ {1, , u} v−1, 1≤ w ≤ v,

S v :{1, , u} v −→ P{1, , u} v−1

,

T v : P

{1, , u} v

−→ P{1, , u} v−1

1.1

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S v,w i1, , i v : i1, , i w−1, i w1, , i v , 1 ≤ w ≤ v,

S v i1, , i v v

w1

{S v,w i1, , i v },

T v I 

⎧

⎪

i1, ,i v ∈I

S v i1, , i v , I / φ,

1.2

Further, introduce the function

via

α v,i i1, , i v : Number of occurrences of i in the sequence i1, , i v . 1.4

For each I ∈ P{1, , u} v, let

α I,i:

i1, ,i v ∈I

α v,i i1, , i v , 1 ≤ i ≤ u. 1.5

It is easy to observe from the construction of the functions S v , S v,w , T v and α v,ithat they do

not depend essentially on u, so we can write for short S v for S u, and so on

H1 The following considerations concern a subset I kof{1, , n} ksatisfying

where n ≥ 1 and k ≥ 2 are fixed integers.

Next, we proceed inductively to define the sets I l ⊂ {1, , n} l k − 1 ≥ l ≥ 1 by

By1.6, I1 {1, , n} and this implies that α I1,i 1 for 1 ≤ i ≤ n From 1.6, again, we have

α I l ,i ≥ 1 k − 1 ≥ l ≥ 1, 1 ≤ i ≤ n.

For every k ≥ l ≥ 2 and for any j1, , j l−1 ∈ I l−1, let

H I l

j1, , j l−1

:i1, , i l , m ∈ I l × {1, , l} | S l,m i1, , i l j1, , j l−1 . 1.8

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Using these sets we define the functions t I k ,l : I l → N k ≥ l ≥ 1 inductively by

t I k ,k i1, , i k : 1, i1, , i k ∈ I k ,

t I k ,l−1

j1, , j l−1

i1, ,i l ,m∈H Ilj1, ,j l−1

t I k ,l i1, , i l . 1.9

Let J be an interval in R, let x : x1, , x n ∈ J n, let p : p1, , p n such that

p i > 0 1 ≤ i ≤ n andn

i1p i 1, and let f : J → R be a convex function For any k ≥ l ≥ 1,

set

A l,l A l,l I k;x; p :

i1, ,i l ∈I l

l

s1

p i s

α I l ,i s

f

l

s1

p i s /α I l ,i s

x i s

l

s1p i s /α I l ,i s

and associate to each k − 1 ≥ l ≥ 1 the number

A k,l A k,l I k;x; p

: k − 11

i1, ,i l ∈I l

t I k ,l i1, , i l

l

s1

p i s

α I k ,i s

f

l

s1

p i s /α I k ,i s

x i s

l

s1p i s /α I k ,i s

We need the following hypotheses

H2 Let x : x1, , x n  and p : p1, , p n be positive n-tuples such thatn

i1p i 1

H3 Let J ⊂ R be an interval, let x : x1, , x n ∈ J n, letp : p1, , p n be a positive

n-tuples such that n

i1p i 1, and let h, g : J → R be continuous and strictly

monotone functions

H4 Let J ⊂ R be an interval, let x : x1, , x n ∈ J n, and letp : p1, , p2 be positive

n-tuples such thatn

p i p i 1 Further, let f : J → R be a convex function.

Assume H1 and H2 The power means of order r ∈ R corresponding to i l :

i1, , i n ∈ I1 l 1, , k are given as

M r

I k ,il :

⎧

⎪

⎛

⎝

l

s1p i s /α I k ,i s x r

i s

l

⎞

⎠

1/r

, r / 0,

l

s1

x p i is /α Ik,is

s

1/l

s1p is /α Ik,is

, r 0.

1.12

We also use the means

M r :

⎧

⎪

n

i1

p i x r i

1/r

, r / 0, n

i1

x p i

i , r 0.

1.13

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For γ, η ∈ R, we introduce the mixed symmetric means with positive weights as follows:

M η,γ1 I k , k :

⎧

⎪

⎡

ik i1, ,i k ∈I k

k

s1

p i s

α I k ,i s

M γ I k ,ikη

⎤

⎦

1/η , η / 0,

ik i1, ,i k ∈I k

M γ I k ,ikk1p is /α Ik,is

, η 0,

1.14

and, for k − 1 ≥ l ≥ 1,

M1η,γ I k , l :

⎧

⎪

⎨

⎪

⎩

⎡

k − 1 l

il i1, ,i l ∈I l

t I k ,lil

l

s1

p i s

α I k ,i s

M γ I k ,ilη

⎤

⎦

1/η

, η / 0,

⎡

il i1, ,i l ∈I l

M γ

I k ,ilt Ik,lill

s1p is /α Ik,is

⎤

⎦

1/k−1 l

, η 0.

1.15

We deduce the monotonicity of these means from the following refinement of the discrete Jensen inequality in1

Theorem 1.1 Assume (H1) and (H4) Then,

f

n

i1

p i x i

≤ A k,k ≤ A k,k−1≤ · · · ≤ A k,2 ≤ A k,1n

i1

p i f x i , 1.16

where the numbers A k,l k ≥ l ≥ 1 are defined in 1.10 and 1.11 If f is a concave function, then

the inequalities in1.16 are reversed.

Under the conditions of the previous theorem,

Υ1

x, p, f: Ak,m − A k,l ≥ 0, k ≥ l > m ≥ 1,

Υ2

x, p, f: Ak,l − f

n

i1

p i x i

Corollary 1.2 Assume (H1) and (H2) Let η, γ ∈ R such that η ≤ γ, then

M γ M1

γ,η I k , 1 ≥ · · · ≥ M1

M η M1

η,γ I k , 1 ≤ · · · ≤ M1

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Proof Assume η, γ / 0 To obtain 1.18, we can apply Theorem 1.1to the function fx

x γ/η x > 0 and the n-tuples x η

1, , x η n to get the analogue of 1.16 and to raise the

power 1/γ Equation1.19 can be proved in a similar way by using fx x η/γ x > 0

andx γ

1, , x γ n and raising the power 1/η.

When η 0 or γ 0, we get the required results by taking limit.

AssumeH1 and H3 Then, we define the quasiarithmetic means with respect to

1.10 and 1.11 as follows:

M1h,g I k , k : h−1

⎛

i1, ,i k ∈I k

k

s1

p i s

α I k ,i s

h ◦ g−1

k

s1

g x i s

k

⎞

and, for k − 1 ≥ l ≥ 1,

M1h,g I k , l h−1

⎛

k − 1 l

il i1, ,i l ∈I l

t I k ,l

ill

s1

p i s

α I k ,i s

h ◦ g−1

l

s1

g x i s

l

⎞

⎠.

1.21 The monotonicity of these generalized means is obtained in the next corollary

Corollary 1.3 Assume (H1) and (H3) For a continuous and strictly monotone function q : J → R,

one defines

M q: q−1

n

i1

p i q x i

Then,

M h M1

h,g I k , 1 ≥ · · · ≥ M1

if either h ◦ g−1is convex and h is increasing or h ◦ g−1is concave and h is decreasing,

M g M1

g,h I k , 1 ≤ · · · ≤ M1

if either g ◦ h−1is convex and g is decreasing or g ◦ h−1is concave and g is increasing.

Proof First, we can apply Theorem 1.1 to the function h ◦ g−1 and the n-tuples

gx1, , gx n , then we can apply h−1 to the inequality coming from 1.16 This gives

1.23 A similar argument gives 1.24: g ◦ h−1, hx1, , hx n and g−1can be used Throughout Examples 1.4-1.5, 1.9–1.12, which are based on examples in 1, the conditionsH2, in the mixed symmetric means, and H3, in the quasiarithmetic means, will

be assumed

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Example 1.4 Suppose

I2:i1, i2 ∈ {1, , n}2| i1|i2

where i1|i2means that i1divides i2 Since i|i i 1, , n, therefore 1.6 holds We note that

α I2,in

i

wheren/i is the largest positive integer not greater than n/i, and di means the number of positive divisors of i Then,1.14 gives for η, γ ∈ R

M η,γ1 I2, 2

⎧

⎪

⎨

⎪

⎩

⎡

i2i1,i2∈I2

2

s1

p is

n/i s di s

M γ

I2,ikn

⎤

⎦

1/n

, η / 0,

i2i1,i2∈I2

M γ

I2,i2 2

1p is / n/i s di s

η / 0,

1.27

while1.20 gives

M1h,g I2, 2 h−1

⎛

i1,i2∈I2

2

s1

p i s

n/i s di s

h ◦ g−1

2

s1

p i s / n/i s di sg x i s

2

s1

p i s / n/i s di s

⎞

⎠.

1.28

AssumeH4 holds, and consider the set I2inExample 1.4 Then,Theorem 1.1implies that

f

n

r1

p r x r

i1,i2∈I2

2

s1

p i s

n/i s di s

f

2

s1

p i s / n/i s di sx i s

2

s1

≤n

r1

p r f x r ,

1.29 and thus

Υ3

x, p, f:

i1,i2∈I2

2

s1

p i s

n/i s di s

f

2

s1

2

s1

−f

n

r1

p r x r

≥0,

Υ4

x, p, f:n

r1

p r f x r −

i1,i2∈I2

2

s1

p i s

n/i s di s

f

2

s1

2

s1

≥ 0.

1.30

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Example 1.5 Let c i ≥ 1 be an integer i 1, , n, let k : n

i1c i , and also let I k P c1, ,c n

k

consist of all sequences i1, , i k in which the number of occurrences of i ∈ {1, , n} is

c i i 1, , n Obviously, 1.6 holds, and, by simple calculations, we have

I k−1n

i1

P c1, ,c i−1,c i −1,c i1, ,c n

k−1 , α I k ,i k!

c1!· · · c n!c i , i 1, , n. 1.31

Moreover, t I k ,k−1i1, , i k−1 k for

i1, , i k−1 ∈ P c1, ,c i−1,c i −1,c i1, ,c n

Under the above settings,1.15 can be written as

M1η,γ I k , k− 1

⎧

⎪

⎡

⎣ 1

k− 1

n

i1

c i − p i

n

r1p r x γ r −p i /c i

x γ i

1−p i /c i

η/γ⎤

⎦

1/η

, η / 0, γ / 0,

⎛

⎝n

i1

n

r1p r x γ r −p i /c i

x γ i

1−p i /c i

c i −p i /γ⎞

⎠

1/k−1

, γ / 0, η 0,

n

i1

x i −p i

n

i1

x r p r

1.33 while1.21 becomes

M h,g1 I k , k − 1 h−1

1

k− 1

n

i1

c i − p i

h ◦ g−1

n

r1p r g x r −p i /c i

g x i

1−p i /c i

. 1.34

AssumeH4 holds, and consider the set I kinExample 1.5 Then,Theorem 1.1yields that

A k,k−1 1

k− 1

n

i1

c i − p i

f

n

r1p r x r−p i /c i

x i

1−p i /c i

,

f

n

r1

p r x r

≤ A k,k−1≤n

r1

p r f x r .

1.35

This shows that

Υ5

x, p, f: Ak,k−1− f

n

r1

p r x r

≥ 0,

Υ6

x, p, f:n

r1

p r f x r − A k,k−1≥ 0.

1.36

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The following result is also given in1.

Theorem 1.6 Assume (H1) and (H4), and suppose |H I I j1, , j l−1| β l−1for any j1, , j l−1 ∈

I l−1 k ≥ l ≥ 2 Then,

A k,l A l,l n

l |I l|

i1, ,i l ∈I l

l

s1

p i s

f

l

s1p i s x i s

l

s1p i s

and thus

f

n

r1

p r x r

≤ A k,k ≤ A k −1,k−1 ≤ · · · ≤ A 2,2 ≤ A 1,1n

r1

p r f x r . 1.38

If f is a concave function then the inequalities1.38 are reversed.

Under the conditions of the previous theorem, we have, from1.38, that

Υ7

x, p, f: Am,m − A l,l ≥ 0, k ≥ l > m ≥ 1,

Υ8

x, p, f: Al,l − f

n

r1

p r x r

AssumeH1 and H2, and suppose |H I I j1, , j l−1| β l−1for anyj1, , j l−1 ∈ I l−1k ≥

l ≥ 2 In this case, the power means of order r ∈ R corresponding to i l : i1, , i l ∈ I l l

1, , k has the form

M r

I l ,il

 M r

I k ,il

⎧

⎪

⎛

⎝

l

s1p i s x r

i s

l

s1p i s

⎞

⎠

1/r

, r / 0,

l

s1

x i s p is

1/l

s1p is

, r 0.

1.40

Now, for γ, η ∈ R and k ≥ l ≥ 1, we introduce the mixed symmetric means with positive

weights related to1.37 as follows:

M2

η,γ I l :

⎧

⎪

⎡

⎣ n

l |I l|

il i1, ,i l ∈I l

l

s1

p i s

M γ

I l ,ilη

⎤

⎦

1/η

, η / 0,

⎡

il i1, ,i l ∈I l

M γ

I l ,ill

s1p is

⎤

⎦

n/l |I l|

, η 0.

1.41

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Corollary 1.7 Assume (H1) and (H2), and suppose |H I I j1, , j l−1| β l−1for any j1, , j l−1 ∈

I l−1k ≥ l ≥ 2 Let η, γ ∈ R such that η ≤ γ Then,

M γ M2

γ,η I1 ≥ · · · ≥ M2

γ,η I k ≥ M η ,

M η M2

η,γ I1 ≤ · · · ≤ M2

η,γ I k ≤ M γ

1.42

Proof The proof comes fromCorollary 1.2

AssumeH1 and H3, and suppose |H I I j1, , j l−1| β l−1 for any j1, , j l−1 ∈

I l−1 k ≥ l ≥ 2 We define for k ≥ l ≥ 1 the quasiarithmetic means with respect to 1.37 as follows:

M2h,g I l : h−1

⎛

⎝ n

l |I l|

i1, ,i l ∈I l

l

s1

p i s

h ◦ g−1l

s1p i s g x i s

l

s1p i s

⎞

Corollary 1.8 Assume (H1) and (H3), and suppose |H I I j1, , j l−1| β l−1for any j1, , j l−1 ∈

I l−1 k ≥ l ≥ 2 Then,

M h M2

h,g I1 ≥ · · · ≥ M2

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

M g M2

g,h I1 ≤ · · · ≤ M2

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

Proof The proof is a consequence ofCorollary 1.3

Example 1.9 If we set

I k:i1, , i k ∈ {1, , n} k | i1< · · · < i k

then α I n ,i 1 i 1, , n, that is, 1.6 is satisfied for k n It comes easily that T k I k

I k−1k 2, , n, |I k| n

k k 1, , n, and for each k 2, , n

H I

k

j1, , j k−1 n − k − 1, j1, , j k−1

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In this case,1.41 becomes for n ≥ k ≥ 1

M2

η,γ I k

⎧

⎪

⎨

⎪

⎩

⎡

n−1

k−1

1≤i 1< ···<i k ≤n

k

s1

p i s

M γ I k ,ikη

⎤

⎥

1/η

, η / 0,

!

1≤i 1< ···<i k ≤n

M γ I k ,ikk1p is

"1/#n−1

k−1

$

, η 0,

1.48

and1.43 has the form

M2

h,g I k h−1

1

n−1

k−1

1≤i 1< ···<i k ≤n

k

s1

p i s

h ◦ g−1k

s1p i s g x i s

k

s1p i s

Equation1.48 is a weighted mixed symmetric mean and 1.49 is a generalized mean, as given in2 Therefore, Corollaries1.7and1.8are more general than the Corollaries 1.2 and 1.3 given in2

AssumeH4 holds, and consider the set I kinExample 1.9 Then,Theorem 1.6shows that

A k,k n1−1

k−1

1≤i 1< ···<i k ≤n

k

s1

p i s

f

k

s1p i s x i s

k

s1p i s

, k 1, , n,

f

n

r1

p r x r

 A n,n ≤ A n −1,n−1 ≤ · · · ≤ A 1,1n

r1

p r f x r .

1.50

Thus, we have

Υ9

x, p, f: Am,m − A l,l ≥ 0, n ≥ l > m ≥ 1. 1.51

Example 1.10 If we set

I k:i1, , i k ∈ {1, , n} k | i1≤ · · · ≤ i k

then α I k ,i ≥ 1 i 1, , n and thus 1.6 is satisfied It is easy to see that T k I k I k−1 k 

2, , |I k| n k−1

k

k 1, , and for each l 2, , k

H I

l

j1, , j l−1 n, j1, , j l−1

Trang 11

Under these settings1.41 becomes

M2η,γ I k

⎧

⎪

⎨

⎪

⎩

⎡

n k−1

k−1

1≤i 1≤···≤i k ≤n

k

s1

p i s

M γ I k ,ikη

⎤

⎥

1/η

, η / 0,

1≤i 1≤···≤i k ≤n

M γ

I k ,ikk

1p is"1/

#n k−1

k−1

$

, η 0,

1.54

and1.43 has the form

M2h,g I k h−1

1

n k−1

k−1

1≤i 1≤···≤i k ≤n

k

s1

p i s

h ◦ g−1k

s1p i s g x i s

k

s1p i s

Equation1.54 represents weighted mixed symmetric means, and 1.55 defines generalized means, as given in 2 Therefore, Corollaries 1.7 and 1.8 are more general than the Corollaries 1.9 and 1.10 given in2

AssumeH4 holds, and consider the set I k in Example 1.10 Then, it follows from

f

n

r1

p r x r

≤ · · · ≤ A k,k ≤ · · · ≤ A 1,1n

r1

where

A k,k n k−11

k−1

1≤i 1≤···≤i k ≤n

k

s1

p i s

f

k

s1p i s x i s

k

s1p i s

This yields that

Υ10

x, p, f A k,k − A l,l ≥ 0, l > k ≥ 1,

Υ11

x, p, f: Ak,k − f

n

r1

p r x r

Example 1.11 We set

Then, α I k ,i ≥ 1 i 1, , n, hence 1.6 holds It is not hard to see that T k I k I k−1 k 

2, , |I k | n k k 1, , and for each l 2, , k,

H I

l

j1, , j l−1 n l ,

j1, , j l−1

is convex and g is decreasing or g ◦ h

is concave and g is increasing.

Proof The proof is a consequence of< /i>Corollary 1.3

Example... γ

1.42

Proof The proof comes fromCorollary 1.2

AssumeH1 and H3, and suppose |H I I j1,... convex and g is decreasing or g ◦ h−1is concave and g is increasing.

Proof First, we can apply Theorem 1.1 to the function h ◦ g−1 and

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