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We will show that this new Cauchy’s mean is monotonic, that is, the following result.. We will prove in Section 3 a comparison theorem for these means.. We need the following lemmas for

Volume 2008, Article ID 163202, 10 pages

doi:10.1155/2008/163202

Research Article

New Means of Cauchy’s Type

Matloob Anwar 1 and J Pe ˇcari ´c 1, 2

1 Abdus Salam School of Mathematical Sciences, GC University, Lahore Gulberg 54660, Pakistan

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

Correspondence should be addressed to Matloob Anwar, matloob t@yahoo.com

Received 30 December 2007; Accepted 7 April 2008

Recommended by Wing-Sum Cheung

We will introduce new means of Cauchy’s type M s r,l f, μ defined, for example, as M s

r,l f, μ 

ll − s/rr − sM r f, μ − M r

s f, μ/M l

l f, μ − M l

s f, μ 1/r−l , in the case when l /  r / s, l, r / 0.

We will show that this new Cauchy’s mean is monotonic, that is, the following result Theorem Let

t, r, u, v∈R, such that t ≤ v, r ≤ u Then for M s

r,l f, μ, one has M s

t,r ≤ M s v,u We will also give some related comparison results.

Copyright q 2008 M Anwar and J Peˇcari´c 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

Let Ω be a convex set equipped with a probability measure μ Then for a strictly monotonic continuous function f, the integral power mean of order r ∈ R is defined as follows:

M rf, μ 

⎧

⎪

⎨

⎪

⎪



Ω



f u r

dμ u 1/r , r /  0,

exp



Ωlog

f u dμ u , r  0.

1.1

Throughout our present investigation, we tacitly assume, without further comment, that all the integrals involved in our results exist

The following inequality for differences of power means was obtained see 1, Remark 8:

r l r − s l − s m ≤ M r r f, μ − M r

s f, μ

M l l f, μ − M l sf, μ

≤ r l r − s l − s M, 1.2

where r, l, s ∈ R, l / r / s, r, l / 0 and where m and M are, respectively, the minimum and the maximum values of the function x r −l on the image of f u u ∈ Ω.

Let us note that1.2 was obtained as consequence of the following result see, e.g., 1, Corollary 1

Theorem 1.1 Let r, s, l ∈ R, and let Ω be a convex set equipped with a probability measure μ Then,

M r r f, μ − M r

s f, μ

M l

l f, μ − M l sf, μ 

r r − s

l l − s η r −l 1.3

for some η in the image of f u u ∈ Ω, provided that the denominator on the left-hand side of 1.3 is

non-zero.

We can also note that from1.3 we can get the following form of 1.2:

inf

u∈Ωf u ≤



l l − s

r r − s

M r r f, μ − M r

s f, μ

M l l f, μ − M l sf, μ

1/r−l

≤ sup

u∈Ωf u, 1.4

where r, l, s ∈ R, r / l / s, r, l / 0 Moreover, 1.4 suggests introducing a new mean of Cauchy type We will prove in Section 3 a comparison theorem for these means Finally we will, in

2 New Cauchy’s mean

From1.4, we can define a new mean M s

r,las follows:

M s r,l f, μ 



l l − s

r r − s

M r r f, μ − M r

s f, μ

M l

l f, μ − M l

s f, μ

1/r−l

, l /  r / s, l, r / 0. 2.1 Now by taking liml→0M s r,l f, μ, we will get

M s r,0 f, μ  M s

0,r f, μ  lim

l→0 M r,l s f, μ



s M r

r f, μ − M r

s f, μ

r r − s log Msf, μ − log M0f, μ

1/r

, r /  s, r, s / 0.

2.2

Now by taking limr →s M s r,l f, μ, we will get

lim

r →s M s r,l f, μ  M s

s,l f, μM s

l,s f, μ





l l − s

s

f u s log f udμu−M s

s f, μ log Ms f, μ

M l

l f, μ−M l sf, μ

1/s−l

, l /  s, l, s / 0.

2.3

By similar way, we can calculate all the cases for r, s, l ∈ R Finally, we get the following

definition of M s

r,l f, μ:

M s r,l f, μ 

l l − s

r r − s

M r

r f, μ − M r

s f, μ

M l l f, μ − M l sf, μ

1/r−l

, l /  r / s, l, r / 0;

M s r,0 f, μ  M s

0,r f, μ 

s M r r f, μ − M r

s f, μ

r r − s log Msf, μ − log M0f, μ

1/r

, r /  s, r, s / 0;

M s s,l f, μ  M s

l,s f, μ 

l l − s

s



f u s log f udμu − M s

s f, μ log Ms f, μ

M l l f, μ − M l sf, μ

1/s−l

,

l /  s, l, s / 0;

M s s,0 f, μ  M s

0,s f, μ 



f u s log f udμu − M s

s f, μ log Ms f, μ

log M sf, μ − log M0f, μ

1/s

, s / 0;

M0r,l f, μ 

l2

M r

r f, μ − M r

0f, μ

r2

M l l f, μ − M l

0f, μ

1/r−l

, l, r / 0;

M0r,0 f, μ  M0

0,r f, μ 

2 M r

r f, μ − M r

0f, μ

r2 M2

2log f, μ − M2

1log f, μ

1/r

, r / 0;

M s t,t exp

− 2t − s

t t − s



f t log fdμu − M t

s f, μ log Msf, μ

M t t f, μ − M t sf, μ



, t /  s;

M0t,t exp

−2

t



f t log fdμu − M t

0f, μ log M0f, μ

M t t f, μ − M t

0f, μ



, t / 0;

M00,0  exp

1 3



log f3dμ u −log M0f, μ 3



log f2dμ u −log M0f, μ 2



,

M s s,s exp

− 1

s



f s log f2dμ u − M s

s f, μlog Msf, μ 2

2 

f s log fdμu −M s f, μ log Msf, μ



, s / 0;

M s 0,0  exp

1

s



log f2dμ u −log M s f, μ 2

2 

log fdμu − log Msf, μ



, s /  0.

2.4

3 Monotonicity of new means

In this section, we will prove the monotonicity of 2.4 We need the following lemmas for log-convex function

Lemma 3.1 Let f be log-convex function and if x1≤ y1, x2≤ y2, x1/  x2, y1/  y2, then the following inequality is valid:

fx

2

f

x1

1/x2−x1 

≤

fy

2

f

y1

1/y2−y1 

Proof In2, page 3 we have the following result for convex function f, with x1 ≤ y1, x2 ≤

y2, x1/  x2, y1/  y2:

f

x2

− fx1

x2− x1 ≤ f



y2

− fy1

y2− y1

Putting f  log f, we get

log

fx

2

f

x1

1/x2−x1 

≤ log

fy

2

f

y1

1/y2−y1 

after applying exponential function we get3.1

The following two lemmas are provedfor functionals in 3 Theorem 4 and Lemma 2,

Lemma 3.2 Let us consider Λ t defined as

Λtg, μ 

⎧

⎪

⎪

⎪

⎪

⎪

⎪

M t t g, μ − M t

1g, μ

t t − 1 , t /  0, 1;

log M1g, μ − log M t

0g, μ, t 0;



g log gμ − M0g, μ log M0g, μ, t  1.

3.4

Then,Λtis a log-convex function.

Lemma 3.3 Let us consider Λ t defined as

Λt 

⎧

⎪

⎨

⎪

1

t2



M t t f, μ − M t

0f, μ , t / 0;

1 2



M22log f, μ − M2

1log f, μ , t  0.

3.5

Then,Λtis a log-convex function.

Theorem 3.4 Let t, r, u, v ∈ R, such that, t ≤ v, r ≤ u Then for 2.4, we have

M s t,r ≤ M s

Case 1 s / 0 Let us consider Λt defined as inLemma 3.2.Λtis a continuous and log-convex

Λt Λr

1tưr

≤

Λv Λu

1/vưu

For s > 0 by substituting g  f s , t  t/s, r  r/s, u  u/s, v  v/s ∈ R, such that, t/s ≤

v/s, r/s ≤ u/s, t / r, v / u, in 3.4, we get

Λt,sf, μ 

⎧

⎪

⎪

⎨

⎪

⎪

⎪

s2

t 1 ư s M t t f, μ ư M t

s f, μ, t /  0, s;

s

log Msf, μ ư log M0f, μ , t 0;

s



f s log f ư M s

0f, μ log M0f, μ , t  s.

3.8

And3.7 becomes

 Λt,s Λr,s

1tưr

≤Λv,s Λu,s

1/vưu

From3.9, we get our required result

Now when s < 0 by substituting g  f s , t  t/s, r  r/s, u  u/s, v  v/s ∈ R, such that, v/s ≤ t/s, u/s ≤ r/s, t / r, v / u, in 3.4 we get 3.8

And3.7 becomes

Λv,s Λu,s

s/ vưu

≤ Λt,s Λr,s

s/ tưr

Now s < 0, from3.10, by raising power ưs, we get

 Λt,s Λr,s

1/tưr

≤Λv,s Λu,s

1/vưu

From3.11, we get our required result

Case 2 s  0 In this case, we can get our result by taking limit s→0 in 3.8 and also in this case we can considerΛtdefined as inLemma 3.3

Λt is log-convex function So, Lemma 3.1 implies that for t, r, u, v ∈ R, such that, t ≤

v, r ≤ u, t / r, v / u, we have

Λt Λr

1/tưr

≤

Λv Λu

1/vưu

Therefore, we have for t, r, u, v ∈ R, such that, t ≤ v, r ≤ u, t / r, v / u:

M0t,r ≤ M0

which completes the proof

4 Further consequences and applications

In this section, we will represent the various applications of our previous definition of a new Cauchy mean and monotonicity of this above defined a new Cauchy mean

4.1 Tobey and Stolarsky-Tobey means

Let En−1represent then − 1-dimensional Euclidean simplex given by

E n−1





u1, u2, , u n−1

: ui ≥ 0, 1 ≤ i ≤ n − 1, n−1

i1

u i≤ 1



, 4.1

and set un  1 −n−1

i1u i Moreover, with u  u1, , u n, let μu be a probability measure on

E n−1 The power mean of order p p ∈ R of the positive n-tuple x  x1, , x n ∈ R n

, with the

weights u  u1, , u n, is defined by

M p x, μ 

⎧

⎪

⎪

⎪

⎪

⎩

n



i1

u i x p i

1/p

, p / 0;

n



i1

x u i

i , p  0.

4.2

Then, the Tobey mean Lp,rx; μ is defined as follows:

L p,r x; μ  MrM px, μ; μ , 4.3

where Mr g, μ denotes the integral power mean, in which Ω is now the n − 1-dimensional Euclidean simplex En−1 We note that, since M px, μ is a mean we have min{xi} ≤ Mpx, μ ≤

max{xi} Now setting fx, μ  Mpx, μ in 2.4 we get

Γs

p,r,l x, μ 

l l − s

r r − s

L r p,r x, μ − L r

p,s x, μ

L l p,l x, μ − L l

p,s x, μ

1/r−l

, l /  r / s, l, r / 0;

Γs

p,r,0 x, μ  Γ s

p,0,r x, μ 

s L r p,r x, μ − L r

p,s x, μ

r r − s log Lp,sx, μ − log Lp,0x, μ

1/r

, r /  s, r, s / 0;

Γs

p,s,l x, μ  Γ s

p,l,s x, μ 

l l − s

s



M px, μ s log dμu − L s

p,s x, μ log Lp,sx, μ

L l p,l x, μ − L l p,sx, μ

1/s−l

,

l /  s, l, s / 0;

Γs

p,s,0 x, μΓ s

p,0,s x, μ



M px, μ s log Mpx, μdμu−L s

p,s x, μ log Lp,sx, μ

log Lp,sx, μ−log Lp,0x, μ

1/s

, s/0;

p,r,l x, μ 

l2

L r p,r x, μ − L r

p,0 x, μ

r2

L l p,l x, μ − L l

p,0 x, μ

1/r−l

, l, r / 0;

Γ0

p,r,0 x, μ  Γ0

p,0,r x, μ 

2 L r p,r x, μ − L r

p,0 x, μ

r2 M2 2



log M px, μ, μ − M2

1



log M px, μ, μ 

1/r

, r / 0;

Γs

p,t,t x, μexp

− 2t − s

t t − s



M px, μ t log Mpx, μdμu−L t

p,s x, μ log Lp,sx, μ

L t p,t x, μ−L t p,sx, μ



, t/ s;

Γ0

p,t,t x, μ  exp

−2

t



M px, μ t log Mpx, μdμu − L t

p,0 x, μ log Lp,0x, μ

L t p,t x, μ − L t

p,0 x, μ



, t / 0;

Γ0

p,0,0 x, μ  exp

1 3

 

log Mpx, μ 3dμ u −log Lp,0x, μ 3

 

log Mpx, μ 2dμ u −log Lp,0x, μ 2



,

Γs

p,s,s x, μexp

−1

s



M px, μ s

log Mpx, μ 2dμ u−L s

p,s x, μlog Lp,sx, μ 2

2 

M px, μ s log Mpx, μdμu−L s p,sx, μ log Lp,sx, μ



, s / 0;

Γs

p,0,0 x, μ  exp

1

s

 

log Mpx, μ 2dμ u −log Lp,sx, μ 2

2 

log M px, μdμu − log Ms x, μ



, s /  0.

4.4

Theorem 4.1 Let t, r, u, v ∈ R, such that, t < v, r < u Then for 4.4, we have

Γs p,t,r ≤ Γs

Proof It is a simple consequence ofTheorem 3.4

Peˇcari´c and ˇSimi´csee 5, Definition 1 introduced the Stolarsky-Tobey mean εp,qx, μ defined by

ε p,qx, μ  Lp,q −p x, ν  Mq −pM px, μ; μ , 4.6

where Lp,rx, ν is the Tobey mean already introduced above.

For the Stolarsky-Tobey mean and2.4, we get the following:

Υs

p,r,l x, μ 

l l − s

r r − s

ε r p,p r x, μ − ε r

p,p s x, μ

ε l p,p l x, μ − ε l

p,p s x, μ

1/r−l

, l /  r / s, l, r / 0;

Υs

p,r,0 x, μ  Υ s

p,0,r x, μ 

s ε r p,p r x, μ − ε r

p,p s x, μ

r r − s log εp,p s x, μ − log εp,p x, μ

1/r

, r /  s, r, s / 0;

p,s,l x, μ  Υ s

p,l,s x, μ 

l l − s

s



M px, μ s log dμu − ε s

p,p s x, μ log εp,p s x, μ

ε l p,p l x, μ − ε l

p,p s x, μ

1/s−l

,

l /  s, l, s / 0;

Υs

p,s,0 x, μ  Υ s

p,0,s x, μ 



M p x, μ s log M p x, μdμu − ε s

p,p s x, μ log εp,p s x, μ

log ε p,p s x, μ − log εp,px, μ

1/s

,

s / 0;

Υ0

p,r,l x, μ 

l2

ε r p,p r x, μ − ε r

p,p x, μ

r2

ε l p,p l x, μ − ε l p,px, μ

1/r−l

, l, r / 0;

Υ0

p,r,0 x, μ  Υ0

p,0,r x, μ 

2εr p,p r x, μ − ε r

p,p x, μ

r2 M2 2



log Mpx, μ, μ − M2

1



log Mpx, μ, μ 

1/r

, r / 0;

Υs

p,t,t x, μ  exp

− 2t − s

t t − s



M px, μ t log Mp x, μdμu − M t

s log εp.p s x, μ

ε p,p t t x, μ − ε t

p,p s x, μ



, t /  s;

Υ0

p,t,t x, μ  exp

− 2

t



M px, μ t log Mpx, μdμu − ε t

P,p x, μ log εp,px, μ

ε t p,p t x, μ − ε t p,px, μ



, t / 0;

Υ0

p,0,0 x, μ  exp

1 3

 

log Mpx, μ 3dμ u −log εp,px, μ 3

 

log Mpx, μ 2dμ u −log εp,px, μ 2



,

Υs

p,s,s x, μ  exp

− 1

s



M px, μ s

log Mpx, μ 2dμ u − ε s

p,p s x, μlog εp.p s x, μ 2

2 

M px, μ s log Mpx, μdμu −ε s p.p s x, μ log εp,p s x, μ



,

s / 0;

Υs

p,0,0 x, μ  exp

1

s

 

log Mpx, μ 2dμ u −log εp,p s x, μ 2

2 

log Mpx, μdμu − log εp.p s x, μ



, s /  0.

4.7

Theorem 4.2 Let t, r, u, v ∈ R, such that, t < v, r < u Then for 4.7, we have

Υs p,t,r ≤ Υs

Proof It is a simple consequence ofTheorem 3.4

4.2 The complete symmetric mean

The rth complete symmetric polynomial meanthe complete symmetric mean of the positive

real n-tuple x is defined bysee 6, pages 332,341

Q n r x q r n x1/r 



c r n x

n r−1

r 

1/r

where c0n x  1 and c r n x n

j1n

i1x i i j and the sum is taken over all n r−1

r  nonnegative

integer n-tuples i1, , i n withn

j1i j  r r / 0 The complete symmetric polynomial mean

can also be written in an integral form as follows:

Q r n 



E n−1

n

i1

x i u i

r

dμ u

1/r

where μ represents a probability measure such that dμu  n−1!du1· · · dun−1 We can see this

as a special case of the integral power mean Mrf, μ, where fu n

i1x i u i , μ is a probability

measure as above, andΩ is the above defined n−1-dimensional simplex En−1 Thus from2.4,

we have the following result:

Θs

n,r,l x, μ 

l l − s

r r − s



Q r n r

x, μ −Q s n r

x, μ



Q l n

l

x, μ −Q s n

l

x, μ

1/r−l

, l /  r / s, l, r ∈ N. 4.11

A simple consequence ofTheorem 3.4is the following result

Theorem 4.3 Let t, r, u, v ∈ N, such that, t < v, r < u Then for 4.11, we have

Θs n,t,r ≤ Θs

4.3 Whiteley means

Let x be a positive real n-tuple, s ∈ R s / 0 and r ∈ N Then, the sth function of degree r is

defined by the following generating functionsee 6, pages 341–344:

∞



r0

t r,s n xt r 

⎧

⎪

⎪

⎪

⎪

n



i1



1 xi t s

, s > 0,

n



i1



1− xi t s

, s < 0.

4.13

The Whiteley mean is now defined by

Wr,s n x w n r,s x1/r

⎧

⎪

⎪

⎨

⎪

⎪

⎩

t r,s n x

nr

s 

1/r

, s > 0,

t r,s n x

−1rnr

s 

1/r

, s < 0.

4.14

For s < 0, the Whiteley mean can be further generalized if we slightly change the definition of

t r,s n x and define h r,σ n x as follows:

∞



r0

h r,σ n xt r n

i1

1



1− xi t σ i , 4.15

where σ  σ1, , σ n; σ ∈ R ; i  1, , n The following generalization of the Whiteley mean for s < 0 is defined bysee 7, Lemma 2.3

Hr,σ n x 

h r,σ n x

n

i1σ i r−1

r

1/r

If we denote by μ a measure on the simplex Δn−1  {u1, , u n : ui ≥ 0, i  1, , n −

1, n

i1u i≤ 1} such that

dμ u  Γ

n

i1σ i

n

i1Γσ i

n

i1

u σ i−1

i du1· · · dun−1, 4.17

where u n 1 −n−1

i−1, then we have μ as a probability measure and we can also write the mean

Hr,σ n x in integral form as follows:

Hr,σ n x 



Δn−1

n



i1

x i u i

r

dμ u

1/r

Finally, just as we did above in this investigation, we can develop the following analogous definition:

Hs

n,r,l x, μ 

⎛

⎜l l − s

r r − s



Hr,σ n

r

x, μ −Hs,σ n

r

x, μ



Hl,σ n

l

x, μ −Hs,σ n

l

x, μ

⎞

⎟

1/r−l

, l /  r / s, l, r ∈ N 4.19

A simple consequence ofTheorem 3.4is the following result

Theorem 4.4 Let t, r, u, v ∈ N, such that, t < v, r < u Then for 4.19, we have

Hs n,t,r ≤ Hs

References

1 J Peˇcari´c, M R Lipanovi´c, and H M Srivastava, “Some mean-value theorems of the Cauchy type,”

Fractional Calculus & Applied Analysis, vol 9, no 2, pp 143–158, 2006.

2 J Peˇcari´c, F Proschan, and Y L Tong, Convex Functions, Partial Orderings, and Statistical Applications, vol 187 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1992.

3 M Anwar and J Peˇcari´c, “On logarithmic convexity for differences of power means,” to appear in

Mathematical Inequalities & Applications.

4 S Simi´c, “On logarithmic convexity for differences of power means,” Journal of Inequalities and

Applications, vol 2007, Article ID 37359, 8 pages, 2007.

5 J Peˇcari´c and V ˇSimi´c, “Stolarsky-Tobey mean in n variables,” Mathematical Inequalities & Applications,

vol 2, no 3, pp 325–341, 1999.

6 P S Bullen, Handbook of Means and Their Inequalities, vol 560 of Mathematics and Its Applications, Kluwer

Academic Publishers, Dordrecht, The Netherlands, 2003.

7 J Peˇcari´c, I Peri´c, and M R Lipanovi´c, “Generalized Whiteley means and related inequalities,” to

appear in Mathematical Inequalities & Applications.

Proof It is a simple consequence of< /i>Theorem 3.4