We generalize means of Stolarsky type and show the monotonicity of these generalized means.. These classical inequalities have been improved and generalized in a number of ways and appli
Trang 1Volume 2010, Article ID 720615, 15 pages
doi:10.1155/2010/720615
Research Article
Generalization of Stolarsky Type Means
J Peˇcari´c1, 2 and G Roqia2
1 Faculty of Textile Technology, University of Zagreb, Pierottijeva, 6, 10000 Zagreb, Croatia
2 Abdus Salam School of Mathematical Sciences, GC University, Lahore 54000, Pakistan
Correspondence should be addressed to G Roqia,rukiyya@gmail.com
Received 27 April 2010; Revised 10 August 2010; Accepted 15 October 2010
Academic Editor: Paolo E Ricci
Copyrightq 2010 J Peˇcari´c and G Roqia 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
We generalize means of Stolarsky type and show the monotonicity of these generalized means
1 Introduction and Preliminaries
The following double inequality is well known in the literature as the Hermite-Hadamard
H.H integral inequality
f
a b
2
≤ 1
b − a
b
a
f xdx ≤ f a fb
provided that f : a, b → Êis a convex function1, page 137, 2, page 1
This result for convex functions plays an important role in nonlinear analysis These classical inequalities have been improved and generalized in a number of ways and
applied for special means including Stolarsky type, logarithmic, and p-logarithmic means A
generalization of H.H inequalities was obtained in3 5, 2, page 5, and 1, page 143
Theorem 1.1 Let p, q be positive real numbers and a1, a, b, b1be real numbers such that a1≤ a <
b ≤ b1 Then the inequalities
f
pa qb
p q
≤ 1
2y
A y
A −y f xdx ≤ pf a qfb
Trang 2hold for A pa qb/p q, y > 0, and all continuous convex functions
f : a1, b1 → Ê if and only if y ≤ b − a/p qmin
p, q
Remark 1.2 The inequalities given by 1.2 are strict if f is a continuous strictly convex
ona1, b1
If we keep the assumptions as stated in Theorem1.1, we also have1, page 146
1
2y
A y
A −y f xdx − f
pa qb
p q
≤ pf a qfb
p q −
1
2y
A y
A −y f xdx. 1.4
The above inequality is strict, when f is strictly convex continuous function.
Let us define F i : Ca, b → Êfor i 1, 2, 3 by differences of 1.2 and 1.4
F1
f; p, q; a, b, y
pf a qfb
p q −
1
2y
M
m
f xdx,
F2
f; p, q; a, b, y
1
2y
M
m
f xdx − f
pa qb
p q
,
F3
f; p, q; a, b, y
pf a qfb
p q f
pa qb
p q
− 1
y
M
m
f xdx,
1.5
where m A − y, M A y.
Remark 1.3 It is clear from inequalities1.2 and 1.4 that if the conditions of Theorem1.1
are satisfied and f ∈ K2a, b f is continuous convex on a, b, then
F i
f; p, q; a, b, y
≥ 0, for i 1, 2, 3. 1.6 Consider the following means:
E r,t
x, y
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
r
y t − x t
t
y r − x r
1/t−r
, tr t − r / 0,
y r − x r
r
log y − log x
1/r
, r / 0, t 0,
e −1/r x
x r
y y r
1/x r −y r
, t r / 0,
1.7
Trang 3where x, y ∈ 0, ∞ such that x / y and r, t ∈Ê These means are known as Stolarsky means Namely, Stolarsky introduced these means in 1975see 1, page 120 and proved that for
r ≤ u and t ≤ v one can get
E r,t
x, y
≤ E u,v
x, y
for x, y ∈ 0, ∞, x / y. 1.8
Some simple proofs of inequality 1.8 and related results on means of Stolarsky type are given in6
The aim of this paper is to prove the exponential convexity of the functions deduced from1.5 and apply these functions to generalize the means of Stolarsky type, and at last we prove the monotonicity property of these new means
We review some necessary definitions and preliminary results
Definition 1.4see 7 A function f : a, b → Ê is exponentially convex if it is continuous and
n
i,j1
ξ i ξ j f
x i x j
for each n ∈ N and every ξ i∈Ê, i 1, , n such that x i x j ∈ a, b, 1 ≤ i, j ≤ n.
Proposition 1.5 see 7 Let f : a, b → Ê, be a function Then f is exponentially convex if and only if f is continuous and
n
i,j1
ξ i ξ j f
x
i x j
2
for all n ∈ N, ξ i∈Êand x i ∈ a, b, 1 ≤ i ≤ n.
Definition 1.6see 1 A function f : I → Ê
, where I is an interval in Ê, is said to be
log-convex if log f is log-convex, or equivalently if for all x, y ∈ I and all α ∈ 0, 1, one has
f
αx 1 − αy≤ f α xf1−αy
Corollary 1.7 see 7 If f : a, b → is exponentially convex then f is log-convex function.
Trang 4The following lemma is another way to define convex function1, page 2.
Lemma 1.8 If f is a convex on an interval I ⊆Ê, then
f s1s3− s2 fs2s1− s3 fs3s1− s2 ≥ 0 1.12
holds for each s1< s2< s3, where s1, s2, s3∈ I.
In Section 2, we prove the exponential and logarithmic convexity of the functions deduced from1.5 We also prove related mean value theorems of Cauchy type
2 Main Results
The following lemma gives us very important family of convex functions
Lemma 2.1 see 7 Consider a family of functions φ r :0, ∞ → Ê, r∈Êdefined as
φ r x
⎧
⎪
⎪
⎪
⎪
x r
r r − 1 , r / 0, 1,
− log x, r 0,
x log x, r 1.
2.1
Then φ r is convex on 0, ∞ for each r ∈Ê.
Theorem 2.2 Let p, q, a, b, A, and y be positive real numbers such that
a < b, A pa qb
p q , y≤
b − a
p qmin
p, q
,
i r : F i
φ r ; p, q; a, b, y
, i 1, 2, 3,
2.2
where φ r is defined in Lemma 2.1 Then
i matrix
i r j r k /2 n
j,k1is positive semidefinite for each n ∈ N and r1, , r n ∈ Ê; particularly,
det
i
r
j r k
2
p
j,k1≥ 0 for 1 ≤ p ≤ n; 2.3
ii the function r → i r is exponentially convex on ;
Trang 5iii if
i r > 0, then the function r →
i r is a log-convex onÊ and the following inequality holds for r, s, t∈Ê such that r < s < t;
i st −r≤
i rt −s
i ts −r 2.4
Proof. i Consider the function
μ x n
j,k1
u j u k φ r jk x 2.5
for 1≤ p ≤ n, x > 0u j∈Ê, where u j is not identically zero and r jk r j r k /2
μ x n
j,k1
u j u k x r jk−2
⎛
⎝n
j1
u j x rj /2−1
⎞
⎠
2
≥ 0 , x > 0.
2.6
This shows that μ is a convex function for x > 0 By setting f μ in 1.5, respectively and from Remark1.3, we get
n
j,k1
u j u k
pφ r jk a qφ r jk b
p q −
1
2y
M
m
φ r jk xdx
≥ 0,
n
j,k1
u j u k 1
2y
M
m
φ r jk xdx − φ r jk
pa qb
p q
≥ 0,
n
j,k1
u j u k pφ r jk a qφ r jk b
p q φ r jk
pa qb
p q
− 1
y
M
m
φ r jk xdx
≥ 0,
2.7
or equivalently
n
j,k1
u j u k
i
r jk
Trang 6
Therefore the given matrix is a positive semidefinite By using well-known Sylvester criterion,
we have
det
i
r
j r k
2
p
j,k1≥ 0 for each 1 ≤ p ≤ n. 2.9
ii Since limr → l
i r
i l for l 0, 1, it follows that
iis continuous onÊ Therefore,
by Proposition1.5for f
i, we get exponential convexity of
ionÊ
iii Let
i r > 0, then the log-convexity of
i is a simple consequence of Corollary1.7
By setting f log
i , s1 r, s2 s, s3 t in Lemma1.8, we have
t − r log
i s ≤ t − s log
i r s − r
i t, 2.10
which implies2.4
We will use the following lemma in the proof of mean value theorem
Lemma 2.3 see 1, page 4 Let f ∈ C2a, b such that
α ≤ f x ≤ β ∀x ∈ a, b. 2.11
If one considers the functions h1, h2, defined by
h1x αx2
2 − fx,
h2x fx − βx2
2 ,
2.12
then h1and h2are convex on a, b.
Proof Therefore
h1x α − f x ≥ 0,
h2x f x − β ≥ 0, 2.13 that is, h j for j 1, 2 are convex on a, b.
Theorem 2.4 Let p, q, a, b, A, and y be real numbers as given in Theorem 1.1 If f ∈ C2a, b
then there exists ξ ∈ a, b such that
F i
f; p, q; a, b, y
f ξ
2 F
i
x2; p, q; a, b, y
for i 1, 2, 3. 2.14
Trang 7Proof Since f ∈ C2a, b, we can take that α ≤ f ≤ β Now in Remark1.3, replacing f by
h j , j 1, 2 defined in Lemma2.3, we have
F i
h j ; p, q; a, b, y
≥ 0 for j 1, 2. 2.15 This gives
F i
f x; p, q : a, b, y≤ β
2F
i
x2; p, q; a, b, y
, α
2F
i
x2; p, q; a, b, y
≤ F i
f x; p, q; a, b, y.
2.16
Combining2.16 and 14, we get
α
2F
i
x2; p, q; a, b, y
≤ F i
f x; p, q; a, b, y≤ β
2F
i
x2; p, q; a, b, y
. 2.17
By using Remark1.2
F i
x2; p, q; a, b, y
> 0, 2.18
therefore
α≤ 2F i
f x; p, q; a, b, y
F i
x2; p, q; a, b, y ≤ β. 2.19
We get the required result
Theorem 2.5 Let p, q, a, b, A, and y be real numbers as given in Theorem 1.1 If f, g ∈ C2a, b
such that g x do not vanish for any x ∈ a, b, then there exits ξ ∈ a, b such that
F i
f; p, q; a, b, y
F i
g; p, q; a, b, y f ξ
g ξ for i 1, 2, 3. 2.20
Proof Define functions φ i ∈ C2a, b, i 1, 2, 3 by
φ i c i
1g − c i
Trang 8c i1 F i
f; p, q; a, b, y
,
c i2 F i
g; p, q; a, b, y
.
2.22
Then using Theorem2.4for f φ i, we have
0
c i1g ξ
2 − c i
2
f ξ
2
F i
x2; p, q; a, b, y
Using Remark1.2
F i
x2; p, q; a, b, y
> 0, 2.24
therefore
c i1
c i2 f ξ
which is clearly2.20
Corollary 2.6 If p, q, a, b, A, and y are real numbers as defined in Theorem 1.1 then for −∞ < r,
t < ∞, r / t, r / 0, 1 and there exists ξ ∈ a, b such that
ξ r −t t t − 1F i
x r ; p, q; a, b, y
r r − 1F i
x t ; p, q; a, b, y for i 1, 2, 3. 2.26
Remark 2.7 If the inverse of f /g exists, then from2.20 we get
ξ
f
g
−1 F i
f; p, q; a, b, y
F i
g; p, q; a, b, y
for i 1, 2, 3. 2.27
Trang 93 Means of Stolarsky Type
Expression2.27 gives the means We can consider
E i r,t
p, q; a, b, y
F i
φ r ; p, q; a, b, y
F i
φ t ; p, q; a, b, y
1/r−t
, r / t, for i 1, 2, 3 3.1
as a means in the broader sense Moreover we can extend these means in other cases Consider the following functions to cover all continuous extensions of3.1:
1r
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
1
r r − 1
pa r qb r
p q −
M r1− m r1
2yr 1
, r / − 1, 0, 1,
pb qa
2ab
p q −
log M − log m
M
log M− 1− mlog m− 1
2y −p log a q log b
p q , r 0,
pa log a qb log b
p q − Γ, r 1,
2r
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
1
r r − 1
M r1− m r1
2yr 1 −
pa qb
p q
r
, r / − 1, 0, 1, log M − log m
4y − p q
2
log
pa qb
p q
− M
log M− 1− mlog m− 1
Γ −pa qb
p q log
pa qb
p q
3r
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
1
r r − 1
pa r qb r
p q
pa qb
p q
r
−M r1− m r1
y r 1
, r / − 1, 0, 1,
pb qa
2ab
p q
p q
2
pa qb −
log M − log m
M
log M− 1− mlog m− 1
y −p log a q log b
p q − log
pa qb
p q
, r 0,
pa log a qb log b
p q
pa qb
p q
log
pa qb
p q
− 1
− 2Γ, r 1,
3.2 whereΓ M22 log M − 1 − m22 log m − 1/8y.
Trang 10We have
E i
r,t
p, q; a, b, y
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
F i
φ r ; p, q; a, b, y
F i
φ t ; p, q; a, b, y
1/r−t
, r / t,
exp 1− 2r
r r − 1 −
F i
φ0φ r ; p, q; a, b, y
F i
φ r ; p, q; a, b, y
, r t / − 1, 0, 1,
exp 3
2 −F i
φ0φ−1; p, q; a, b, y
F i
φ−1; p, q; a, b, y
, r t −1,
exp 1− F i
φ2
0; p, q; a, b, y
2F i
φ0; p, q; a, b, y
, r t 0,
exp −1 −F i
φ0φ1; p, q; a, b, y
2F i
φ1; p, q; a, b, y
, r t 1,
3.3
for i 1, 2, 3 We will use the following lemma to prove the monotonicity of Stolarsky type
means
Lemma 3.1 Let f be log-convex function, and if r ≤ u, t ≤ v, r / t, u / v, then the following
inequality is valid:
f r
f t
1/r−t
≤
f u
f v
1/u−v
The proof of this Lemma is given in1
Theorem 3.2 Let p, q, a, b, A, and y be real numbers as defined in Theorem 1.1 and let r, t, u, v∈Ê
such that r ≤ u, t ≤ v, then the following inequality is valid:
E i r,t
p, q; a, b, y
≤ E i u,v
p, q; a, b, y
for i 1, 2, 3. 3.5
Proof For a convex function φ, a simple consequence of the definition of convex function is
the following inequality1, page 2:
φ x1 − φx2
x2− x1 ≤ φ
y2
− φy1
y2− y1
, with x1≤ y1, x2≤ y2, x1/ x2, y1/ y2. 3.6
As
i is log-convex we set φr log
i r, x1 r, x2 t, y1 v, y2 u in the above inequality
and get
log
i r − log
i t
r − t ≤
log
i u − log
i v
which is equivalent to3.5 for t / r, u / v By continuity of i,3.5 is valid for t r, u v.
Trang 11Remark 3.3 If we substitute p q 1 and replace r → r − 1 and t → t − 1 in E i
r,t p, q; a, b, y, for i 1, 2, 3, then means of Stolarsky type and related results given in 6 are obtained
4 Generalized Means of Stolarsky Type
By substiting a → a s , b → b s , y → y s , r → r/s, t → t/s, ξ → ξ 1/sin2.26, we get
ξ r −t t t − sF i
x r/s ; p, q; a s , b s , y s
r r − sF i
x t/s ; p, q; a s , b s , y s , s / 0, t / r for i 1, 2, 3. 4.1
It follows that
E i r,t;s
p, q; a s , b s , y s
F i
φ r/s ; p, q; a s , b s , y s
F i
φ t/s ; p, q; a s , b s , y s
1/r−t
, s / 0, t / r for i 1, 2, 3. 4.2
To get all continuous extension of4.2, we consider
A
⎧
⎪
⎪
pa s qb s
p q
1/s
, s / 0,
a p b q1/pq , s 0,
y≤
⎧
⎪
⎨
⎪
⎩
b s − a s
p q min
p, q1/s
, s / 0,
b a
1/pq min{p,q}
, s 0.
4.3
For s / 0, we define
i
s r F i
φ r/s ; p, q; a s , b s , y s
for i 1, 2, 3, 4.4
where {φ r ; r ∈ Ê} is the family of functions defined in Lemma 2.1 Here we
have F i f; p, q; a s , b s , y s defined as
F1
f; p, q; a s , b s , y s
pf a s qfb s
p q −
1
2y s
M s
m s
f xdx,
F2
f; p, q; a s , b s , y s
1
2y s
M s
m s
f xdx − f
pa s qb s
p q
,
F3
f; p, q; a s , b s , y s
fpa s qb s
p q
pf a s qfb s
p q −
1
y s
M s
m s
f xdx,
4.5
where i 1, 2, 3, m s A s − y s , and M s A s y s
Trang 12We have
1
s r
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
s2
r r − s
pa r qb r
p q −
s
r s
M s rs/s − m rs/s s
2y s
, r / − s, 0, s,
pb s qa s
2a s b s
p q −
log M s − log m s
M slog M s− 1− m s
log m s− 1
2y s − s p log a q log b
p q , r 0,
s pa
s log a qb s log b
p q − Γs , r s,
2
s r
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
s2
r r − s
s
r s
M s rs/s − m s rs/s
pa s qb s
p q
r/s
, r / − s, 0, s,
log M s − log m s
4y s − p q
2
pa s qb s
s
M s
log M s− 1− m s
log m s− 1
2y s − s log pa s qb s
p q , r 0,
Γs − s
pa s qb s
p q
s
log
pa s qb s
p q
3
s r
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
s2
r r − s
pa r qb r
p q
pa s qb s
p q
r/s
− s
r s
M rs/s s − m rs/s s
y s
,
r / − s, 0, s;
pb s qa s
2a s b s
p q
p q
2
pa s qb s
s
−log M s − log m s
M s
log M s− 1− m s
log m s− 1
pa s qb s
p q
−s p log a q log b
s pa log a qb log b
p q s
pa s qb s
p q
s
log
pa s qb s
p q
− 2Γs , r s,
4.6
whereΓs M s 2 log M s − 1 − m s 2 log m s − 1/8y s
Trang 13For s 0, we consider a family of convex functions {ψ r : r∈Ê} defined onÊ by
ψ r x
⎧
⎪
⎪
1
r2e rx , r / 0,
1
2x
We have F i f; p, q; log a, log b, log y, i 1, 2, 3 defined as
F1
f; p, q; log a, log b, log y
pf
log a
qflog b
p q −
1
2 log y
log M
log m
f xdx,
F2
f; p, q; log a, log b, log y
1
2 log y
log M
log m
f xdx − f p
log a
qlog b
p q
,
F3
f; p, q; log a, log b, log y
f
p log a q log b
p q
pf
log a
qflog b
p q
− 1
log y
log M
log m
f xdx,
4.8
where log m loga p b q1/pq /y , log M logya p b q1/pq Now for
i
0r F i
ψ r ; p, q; log a, log b, log y
for i 1, 2, 3,
1
0r
⎧
⎪
⎪
⎪
⎪
1
r2
pa r qb r
p q −
a p b qr/ pq
y 2r− 1
2ry r log y
, r / 0,
1 2
p log2a q log2b
p q − log2a p b q1/pq−
1
3log
2y
, r 0,
2
0r
⎧
⎪
⎪
1
r2
a p b qr/ pq
y 2r− 1
2ry r log y − a p b qr/ pq
, r / 0,
1
6log
3
0r
⎧
⎪
⎪
⎪
⎪
1
r2
pa r qb r
p q a p b qr/ pq−a p b qr/ pq
y 2r− 1
ry r log y
, r / 0,
1 2
p log2a q log2b
p q − log2a p b q1/pq−
2
3log
2y
, r 0.
4.9