Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 163202, 10 pages
doi:10.1155/2008/163202
Research Article
New Means of Cauchy’s Type
Matloob Anwar1 and J. Pečarić1, 2
1
2
Abdus Salam School of Mathematical Sciences, GC University, Lahore Gulberg 54660, Pakistan
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
s
s
f, μ defined, for example, as Mr,l
f, μ We will introduce new means of Cauchy’s type Mr,l
r /
s, l, r /
0.
ll − s/rr − sMrr f, μ − Msr f, μ/Mll f, μ − Msl f, μ1/r−l , in the case when l /
We will show that this new Cauchy’s mean is monotonic, that is, the following result. Theorem. Let
s
s
s
f, μ, one has Mt,r
≤ Mv,u
. We will also give some
t, r, u, v ∈ R, such that t ≤ v, r ≤ u. Then for Mr,l
related comparison results.
Copyright q 2008 M. Anwar and J. Pečarić. 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:
Mr f, μ ⎧
1/r
r
⎪
⎪
⎪
dμu
,
fu
⎨
Ω
r/
0,
⎪
⎪
⎪
⎩exp
log fu dμu , r 0.
1.1
Ω
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
r
rr − s m ≤ Mr f, μ − Ms f, μ ≤ rr − s M,
l
ll − s ll − s l
M f, μ − Ms f, μ
l
1.2
2
Journal of Inequalities and Applications
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 xr−l on the image of fu u ∈ Ω.
Let us note that 1.2 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,
Mrr f, μ − Msr f, μ
Mll f, μ
Msl f, μ
−
rr − s r−l
η
ll − s
1.3
for some η in the image of fu u ∈ Ω, provided that the denominator on the left-hand side of 1.3 is
non-zero.
We can also note that from 1.3 we can get the following form of 1.2:
inf fu ≤
u∈Ω
ll − s Mrr f, μ − Msr f, μ
rr − s Ml f, μ − Msl f, μ
1/r−l
≤ sup fu,
1.4
u∈Ω
l
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
Section 4, give some applications.
2. New Cauchy’s mean
s
From 1.4, we can define a new mean Mr,l
as follows:
s
f, μ
Mr,l
ll − s Mrr f, μ − Msr f, μ
rr − s Ml f, μ − Msl f, μ
1/r−l
,
l/
r /
s, l, r /
0.
2.1
l
s
Now by taking lim l→0 Mr,l
f, μ, we will get
s
s
s
Mr,0
f, μ M0,r
f, μ lim Mr,l
f, μ
l→0
1/r
s Mrr f, μ − Msr f, μ
,
rr − s log Ms f, μ − log M0 f, μ
2.2
r/
s, r, s /
0.
s
f, μ, we will get
Now by taking lim r→s Mr,l
s
s
s
lim Mr,l
f, μ Ms,l
f, μ Ml,s
f, μ
r→s
ll − s
s
1/s−l
fus log fudμu−Mss f, μ log Ms f, μ
Mll f, μ−Msl f, μ
,
l/
s, l, s /
0.
2.3
M. Anwar and J. Pečarić
3
By similar way, we can calculate all the cases for r, s, l ∈ R. Finally, we get the following
s
definition of Mr,l
f, μ:
s
Mr,l
f, μ
ll − s Mrr f, μ − Msr f, μ
rr − s Ml f, μ − Msl f, μ
1/r−l
,
l/
r /
s, l, r /
0;
l
s
Mr,0
f, μ
s
M0,r
f, μ
s
Ml,s
f, μ
s
Ms,l
f, μ
1/r
s Mrr f, μ − Msr f, μ
,
rr − s log Ms f, μ − log M0 f, μ
ll − s
s
r/
s, r, s /
0;
fus log fudμu − Mss f, μ log Ms f, μ
1/s−l
,
Mll f, μ − Msl f, μ
l/
s, l, s /
0;
s
Ms,0
f, μ
0
Mr,l
f, μ
s
M0,s
f, μ
fus log fudμu − Mss f, μ log Ms f, μ
log Ms f, μ − log M0 f, μ
2 r
1/r−l
l Mr f, μ − M0r f, μ
,
r 2 Mll f, μ − M0l f, μ
0
f, μ
Mr,0
0
M0,r
f, μ
s
Mt,t
exp
0
Mt,t
0
M0,0
exp
2t − s
−
tt − s
2
− t
exp
s
M0,0
s/
0;
l, r / 0;
1/r
2 Mrr f, μ − M0r f, μ
r 2 M22 log f, μ − M12 log f, μ
,
f t log fdμu − Mst f, μ log Ms f, μ
Mtt f, μ − Mst f, μ
f t log fdμu − M0t f, μ log M0 f, μ
Mtt f, μ − M0t f, μ
r/
0;
,
,
t
/ s;
,
t/
0;
3 3
1 log f dμu − log M0 f, μ
exp
,
3 log f2 dμu − log M0 f, μ2
s
Ms,s
1/s
exp
s
2 f log f2 dμu − Mss f, μ log Ms f, μ
1
− s
,
s 2 f log fdμu − Mss f, μ log Ms f, μ
2 log f2 dμu − log Ms f, μ
1
,
s
2 log fdμu − log Ms f, μ
s
/ 0;
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.
4
Journal of Inequalities and Applications
Lemma 3.1. Let f be log-convex function and if x1 ≤ y1 , x2 ≤ y2 , x1 /
x2 , y 1 /
y2 , then the following
inequality is valid:
1/x2 −x1 1/y2 −y1 f x2
f y2
≤
.
f x1
f y1
3.1
Proof. In 2, page 3 we have the following result for convex function f, with x1 ≤ y1 , x2 ≤
x2 , y 1 /
y2 :
y2 , x1 /
f x2 − f x1
f y2 − f y 1
≤
.
x2 − x1
y2 − y 1
3.2
1/x2 −x1 1/y2 −y1 f x2
f y2
log
≤ log
,
f x1
f y1
3.3
Putting f log f, we get
after applying exponential function we get 3.1.
The following two lemmas are proved for functionals in 3 Theorem 4 and Lemma 2,
for Lemma 3.2 see also 4, Theorem 1.
Lemma 3.2. Let us consider Λt defined as
⎧
⎪
Mtt g, μ − M1t g, μ
⎪
⎪
,
t/
0, 1;
⎪
⎪
⎪
tt − 1
⎪
⎨
t 0;
Λt g, μ log M1 g, μ − log M0t g, μ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩ g log gμ − M0 g, μ log M0 g, μ, t 1.
3.4
Then, Λt is a log-convex function.
Lemma 3.3. Let us consider Λt defined as
⎧
1
⎪
⎪
t/
0;
⎨ 2 Mtt f, μ − M0t f, μ ,
Λt t
⎪
⎪
⎩ 1 M2 log f, μ − M2 log f, μ , t 0.
2
1
2
3.5
Then, Λt is 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
s
s
Mt,r
≤ Mv,u
.
3.6
M. Anwar and J. Pečarić
5
Proof
Case 1 s /
0. Let us consider Λt defined as in Lemma 3.2. Λt is a continuous and log-convex.
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
1t−r
≤
Λv
Λu
1/v−u
3.7
.
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
⎧
s2 t
⎪
⎪
t/
0, s;
Mt f, μ − Mst f, μ ,
⎪
⎪
⎪
t1 − s
⎪
⎨ t 0;
Λt,s f, μ s log Ms f, μ − log M0 f, μ ,
⎪
⎪
⎪ ⎪
⎪
⎪
⎩s f s log f − M0s f, μ log M0 f, μ , t s.
3.8
And 3.7 becomes
Λt,s
Λr,s
1t−r
≤
Λv,s
Λu,s
1/v−u
.
3.9
From 3.9, 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.
And 3.7 becomes
Λv,s
Λu,s
s/v−u
≤
Λt,s
Λr,s
s/t−r
.
3.10
.
3.11
Now s < 0, from 3.10, by raising power −s, we get
Λt,s
Λr,s
1/t−r
≤
Λv,s
Λu,s
1/v−u
From 3.11, 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 consider Λt defined as in Lemma 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/t−r
≤
Λv
Λu
1/v−u
.
3.12
r, v /
u:
Therefore, we have for t, r, u, v ∈ R, such that, t ≤ v, r ≤ u, t /
0
0
Mt,r
≤ Mv,u
,
which completes the proof.
3.13
6
Journal of Inequalities and Applications
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−1 represent the n − 1-dimensional Euclidean simplex given by
En−1 n−1
u1 , u2 , . . . , un−1 : ui ≥ 0, 1 ≤ i ≤ n − 1,
ui ≤ 1 ,
4.1
i1
and set un 1 − n−1
i1 ui . Moreover, with u u1 , . . . , un , let μu be a probability measure on
En−1 . The power mean of order p p ∈ R of the positive n-tuple x x1 , . . . , xn ∈ Rn , with the
weights u u1 , . . . , un , is defined by
Mp x, μ ⎧
1/p
n
⎪
⎪
p
⎪
⎪
ui xi
, p/
0;
⎪
⎨
i1
n
⎪
⎪
ui
⎪
⎪
⎪
⎩ xi ,
4.2
p 0.
i1
Then, the Tobey mean Lp,r x; μ is defined as follows:
Lp,r x; μ Mr Mp x, 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 Mp x, μ is a mean we have min{xi } ≤ Mp x, μ ≤
max{xi }. Now setting fx, μ Mp x, μ in 2.4 we get
Γsp,r,l x, μ
r
r
ll − s Lp,r x, μ − Lp,s x, μ
rr − s Ll x, μ − Llp,s x, μ
1/r−l
,
l
/r / s, l, r / 0;
p,l
1/r
s Lrp,r x, μ − Lrp,s x, μ
, r
/ s, r, s / 0;
rr − s log Lp,s x, μ − log Lp,0 x, μ
1/s−l
s
s
ll − s Mp x, μ log dμu − Lp,s x, μ log Lp,s x, μ
s
s
,
Γp,s,l x, μ Γp,l,s x, μ s
Llp,l x, μ − Llp,s x, μ
Γsp,r,0 x, μ
Γsp,0,r x, μ
Γsp,s,0 x, μ Γsp,0,s x, μ l/
s, l, s /
0;
1/s
Mp x, μs log Mp x, μdμu−Lsp,s x, μ log Lp,s x, μ
, s
/0;
log Lp,s x, μ−log Lp,0 x, μ
M. Anwar and J. Pečarić
1/r−l
2 r
l Lp,r x, μ − Lrp,0 x, μ
0
Γp,r,l x, μ ,
r 2 Llp,l x, μ − Llp,0 x, μ
Γ0p,r,0 x, μ
Γ0p,0,r x, μ
Γsp,t,t x, μ exp
7
l, r / 0;
1/r
2 Lrp,r x, μ − Lrp,0 x, μ
,
r 2 M22 log Mp x, μ, μ − M12 log Mp x, μ, μ
Mp x, μt log Mp x, μdμu−Ltp,s x, μ log Lp,s x, μ
2t − s
−
, t/
s;
tt − s
Ltp,t x, μ−Ltp,s x, μ
Mp x, μt log Mp x, μdμu − Ltp,0 x, μ log Lp,0 x, μ
2
− ,
t
Ltp,t x, μ − Ltp,0 x, μ
Γ0p,t,t x, μ
exp
Γ0p,0,0 x, μ
3
3 1 log Mp x, μ dμu − log Lp,0 x, μ
exp
,
3 log M x, μ2 dμu − log L x, μ2
p
Γsp,s,s x, μ exp
exp
t/
0;
p,0
2
2 Mp x, μs log Mp x, μ dμu−Lsp,s x, μ log Lp,s x, μ
1
− ,
s 2 Mp x, μs log Mp x, μdμu− Lsp,s x, μ log Lp,s x, μ
Γsp,0,0 x, μ
r/
0;
2
2 log Mp x, μ dμu − log Lp,s x, μ
1
,
s
2 log Mp x, μdμu − log Ms x, μ
s
/ 0;
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
Γsp,t,r ≤ Γsp,v,u .
4.5
Proof. It is a simple consequence of Theorem 3.4.
Pečarić and Šimić see 5, Definition 1 introduced the Stolarsky-Tobey mean εp,q x, μ
defined by
εp,q x, μ Lp,q−p x, ν Mq−p Mp x, μ; μ ,
4.6
where Lp,r x, ν is the Tobey mean already introduced above.
For the Stolarsky-Tobey mean and 2.4, we get the following:
s
x, μ
Υp,r,l
r
r
ll − s εp,pr x, μ − εp,ps x, μ
l
rr − s εl x, μ − εp,ps
x, μ
1/r−l
,
l/
r /
s, l, r /
0;
p,pl
s
Υp,r,0
x, μ
s
Υp,0,r
x, μ
r
1/r
r
s εp,pr
x, μ − εp,ps
x, μ
,
rr − s log εp,ps x, μ − log εp,p x, μ
r
/ s, r, s / 0;
8
Journal of Inequalities and Applications
s
Υp,s,l
x, μ
s
Υp,l,s
x, μ
s
s
ll − s Mp x, μ log dμu − εp,ps x, μ log εp,ps x, μ
l
s
εl x, μ − εp,ps
x, μ
1/s−l
,
p,pl
s
s
Υp,s,0
x, μ Υp,0,s
x, μ l/
s, l, s /
0;
1/s
s
Mp x, μs log Mp x, μdμu − εp,ps
x, μ log εp,ps x, μ
log εp,ps x, μ − log εp,p x, μ
,
s/
0;
1/r−l
2 r
r
l εp,pr x, μ − εp,p
x, μ
0
,
Υp,r,l x, μ l
l
r 2 εp,pl
x, μ − εp,p
x, μ
0
Υp,r,0
x, μ
0
Υp,0,r
x, μ
s
x, μ
Υp,t,t
exp
0
x, μ exp
Υp,t,t
0
x, μ
Υp,0,0
l, r / 0;
r
r
2εp,pr
x, μ − εp,p
x, μ
2
2
r 2 M2 log Mp x, μ, μ − M1 log Mp x, μ, μ
1/r
,
Mp x, μt log Mp x, μdμu − Mst log εp.ps x, μ
2t − s
−
,
t
t
tt − s
εp,pt
x, μ − εp,ps
x, μ
t
Mp x, μt log Mp x, μdμu − εP,p
x, μ log εp,p x, μ
2
− ,
t
t
t
εp,pt
x, μ − εp,p
x, μ
r/
0;
t/
s;
t/
0;
3
3 1 log Mp x, μ dμu − log εp,p x, μ
exp
,
3 log M x, μ2 dμu − log ε x, μ2
p
s
Υp,s,s
x, μ exp
p,p
2
2 s
Mp x, μs log Mp x, μ dμu − εp,ps
x, μlog εp.ps x, μ
1
− s
,
s 2 Mp x, μs log Mp x, μdμu − εp.ps
x, μ log εp,ps x, μ
s/
0;
s
Υp,0,0
x, μ
exp
2
2 log Mp x, μ dμu − log εp,ps x, μ
1
,
s
2 log Mp x, μdμu − log εp.ps 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
s
s
≤ Υp,v,u
.
Υp,t,r
Proof. It is a simple consequence of Theorem 3.4.
4.8
M. Anwar and J. Pečarić
9
4.2. The complete symmetric mean
The rth complete symmetric polynomial mean the complete symmetric mean of the positive
real n-tuple x is defined by see 6, pages 332,341
1/r cr x 1/r
r
r
n
Qn x qn x
,
nr−1
r 4.9
ij
0
r
where cn x 1 and cn x nj1 ni1 xi and the sum is taken over all nr−1
r nonnegative
n
integer n-tuples i1 , . . . , in with j1 ij r r 0.
The
complete
symmetric
polynomial
mean
/
can also be written in an integral form as follows:
r
Qn
n
En−1
1/r
r
xi ui
dμu
4.10
,
i1
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 Mr f, μ, where fu ni1 xi ui , μ is a probability
measure as above, and Ω is the above defined n−1-dimensional simplex En−1 . Thus from 2.4,
we have the following result:
Θsn,r,l x, μ
1/r−l
s r
r r
ll − s Qn x, μ − Qn x, μ
,
s l
rr − s l l
Qn x, μ − Qn x, μ
l/
r /
s, l, r ∈ N.
4.11
A simple consequence of Theorem 3.4 is the following result.
Theorem 4.3. Let t, r, u, v ∈ N, such that, t < v, r < u. Then for 4.11, we have
Θsn,t,r ≤ Θsn,v,u .
4.12
4.3. Whiteley means
0 and r ∈ N. Then, the sth function of degree r is
Let x be a positive real n-tuple, s ∈ R s /
defined by the following generating function see 6, pages 341–344:
∞
r,s
tn
xtr r0
⎧ n
s
⎪
⎪
⎪
1 xi t , s > 0,
⎪
⎨
i1
n
⎪
s
⎪
⎪
⎪
1 − xi t , s < 0.
⎩
4.13
i1
The Whiteley mean is now defined by
r,s
Wn
1/r
r,s
x wn x
⎧
1/r
r,s
⎪
tn x
⎪
⎪
⎪
,
s > 0,
⎪
⎨
nrs 1/r
⎪
r,s
⎪
⎪
tn x
⎪
⎪
, s < 0.
⎩
−1r nrs 4.14
10
Journal of Inequalities and Applications
For s < 0, the Whiteley mean can be further generalized if we slightly change the definition of
r,s
r,σ
tn x and define hn x as follows:
∞
r,σ
hn
xtr r0
n
i1
1
σ ,
1 − xi t i
4.15
where σ σ1 , . . . , σn ; σ ∈ R ; i 1, . . . , n. The following generalization of the Whiteley mean
for s < 0 is defined by see 7, Lemma 2.3
r,σ
1/r
hn x
r,σ
Hn x n σ r−1 .
4.16
i1 i
r
If we denote by μ a measure on the simplex Δn−1 {u1 , . . . , un : ui ≥ 0, i 1, . . . , n −
n
1,
i1 ui ≤ 1} such that
n
Γ n σi dμu n i1 uσi i −1 du1 · · · dun−1 ,
4.17
Γ
σ
i i1
i1
where un 1 − n−1
i−1 , then we have μ as a probability measure and we can also write the mean
r,σ
Hn x in integral form as follows:
1/r
r
n
r,σ
Hn x xi ui dμu
.
4.18
Δn−1
i1
Finally, just as we did above in this investigation, we can develop the following analogous
definition:
⎛
⎞1/r−l
r,σ r
s,σ r
x,
μ
−
H
x,
μ
H
n
n
⎜ ll − s
⎟
Hsn,r,l x, μ ⎝
, l/
r ⎠
/ s, l, r ∈ N. 4.19
l
l
rr − s
l,σ
s,σ
Hn
x, μ − Hn
x, μ
A simple consequence of Theorem 3.4 is the following result.
Theorem 4.4. Let t, r, u, v ∈ N, such that, t < v, r < u. Then for 4.19, we have
Hsn,t,r ≤ Hsn,v,u .
4.20
References
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