Document 10942606

advertisement
Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 720615, 15 pages
doi:10.1155/2010/720615
Research Article
Generalization of Stolarsky Type Means
J. Pečarić1, 2 and G. Roqia2
1
2
Faculty of Textile Technology, University of Zagreb, Pierottijeva, 6, 10000 Zagreb, Croatia
Abdus Salam School of Mathematical Sciences, GC University, Lahore 54000, Pakistan
Correspondence should be addressed to G. Roqia, [email protected]
Received 27 April 2010; Revised 10 August 2010; Accepted 15 October 2010
Academic Editor: Paolo E. Ricci
Copyright q 2010 J. Pečarić 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
ab
2
≤
1
b−a
b
fxdx ≤
a
fa fb
,
2
1.1
provided that f : a, b → Ê is a convex function 1, 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 in 3–5, 2, page 5, and 1, page 143.
Theorem 1.1. Let p, q be positive real numbers and a1 , a, b, b1 be real numbers such that a1 ≤ a <
b ≤ b1 . Then the inequalities
f
pa qb
pq
≤
1
2y
Ay
A−y
fxdx ≤
pfa qfb
pq
1.2
2
Journal of Inequalities and Applications
hold 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 q min p, q .
1.3
Remark 1.2. The inequalities given by 1.2 are strict if f is a continuous strictly convex
on a1 , b1 .
If we keep the assumptions as stated in Theorem 1.1, we also have 1, page 146
1
2y
Ay
A−y
pa qb
fxdx − f
pq
pfa qfb
1
−
≤
pq
2y
Ay
fxdx.
1.4
A−y
The above inequality is strict, when f is strictly convex continuous function.
Let us define F i : Ca, b → Ê for i 1, 2, 3 by differences of 1.2 and 1.4
pfa qfb
1 M
F 1 f; p, q; a, b, y fxdx,
−
pq
2y m
pa qb
1 M
fxdx − f
F 2 f; p, q; a, b, y ,
2y m
pq
pfa qfb
pa qb
1 M
3
f
−
fxdx,
F f; p, q; a, b, y pq
pq
y m
1.5
where m A − y, M A y.
Remark 1.3. It is clear from inequalities 1.2 and 1.4 that if the conditions of Theorem 1.1
are satisfied and f ∈ K2 a, 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:
Er,t x, y ⎧ 1/t−r
⎪
r y t − xt
⎪
⎪
⎪ ,
⎪
⎪
⎪
t y r − xr
⎪
⎪
1/r
⎪
⎪
⎪
y r − xr
⎪
⎪
⎪
,
⎪
⎨ r log y − log x
⎪
r 1/xr −yr ⎪
⎪
⎪
⎪
xx
⎪
−1/r
⎪
,
e
⎪
r
⎪
yy
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩√
xy,
trt − r /
0,
r/
0, t 0,
1.7
tr/
0,
t r 0,
Journal of Inequalities and Applications
3
where x, y ∈ 0, ∞ such that x /
y and r, t ∈ Ê. These means are known as Stolarsky means.
Namely, Stolarsky introduced these means in 1975 see 1, page 120 and proved that for
r ≤ u and t ≤ v one can get
Er,t x, y ≤ Eu,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 in 6.
The aim of this paper is to prove the exponential convexity of the functions deduced
from 1.5 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 1.4 see 7. A function f : a, b →
and
n
Ê
is exponentially convex if it is continuous
ξi ξj f xi xj ≥ 0,
1.9
i,j1
for each n ∈ N and every ξi ∈ Ê, i 1, . . . , n such that xi xj ∈ a, b, 1 ≤ i, j ≤ n.
Proposition 1.5 see 7. Let f : a, b →
only if f is continuous and
n
Ê, be a function. Then f is exponentially convex if and
ξi ξj f
i,j1
xi xj
2
≥ 0,
1.10
for all n ∈ N, ξi ∈ Ê and xi ∈ a, b, 1 ≤ i ≤ n.
Definition 1.6 see 1. A function f : I → Ê , where I is an interval in Ê, is said to be logconvex if log f is convex, or equivalently if for all x, y ∈ I and all α ∈ 0, 1, one has
f αx 1 − αy ≤ f α xf 1−α y .
Corollary 1.7 see 7. If f : a, b →
Ê is exponentially convex then f
1.11
is log-convex function.
4
Journal of Inequalities and Applications
The following lemma is another way to define convex function 1, page 2.
Lemma 1.8. If f is a convex on an interval I ⊆ Ê, then
fs1 s3 − s2 fs2 s1 − s3 fs3 s1 − 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 from 1.5. 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 φr : 0, ∞ →
⎧
xr
⎪
⎪
,
⎪
⎪
⎨ rr − 1
φr x − log x,
⎪
⎪
⎪
⎪
⎩
x log x,
Ê, r ∈ Ê defined as
r
/ 0, 1,
r 0,
2.1
r 1.
Then φr 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
pa qb
b−a
, y≤
min p, q ,
pq
pq
i r : F i φr ; p, q; a, b, y , i 1, 2, 3,
a < b,
A
2.2
where φr is defined in Lemma 2.1. Then
i matrix i rj rk /2nj,k1 is positive semidefinite for each n ∈ N and r1 , . . . , rn ∈
particularly,
rj rk p
det i
≥ 0 for 1 ≤ p ≤ n;
2
j,k1
ii the function r → i r is exponentially convex on Ê;
Ê;
2.3
Journal of Inequalities and Applications
5
iii if i r > 0, then the function r → i r is a log-convex on
holds for r, s, t ∈ Ê such that r < s < t;
Proof.
t−r
Ê
and the following inequality
t−s s−r
≤ i r
i t
.
i s
2.4
i Consider the function
μx n
2.5
uj uk φrjk x
j,k1
for 1 ≤ p ≤ n, x > 0uj ∈ Ê, where uj is not identically zero and rjk rj rk /2
n
μ
x uj uk xrjk −2
j,k1
⎛
⎝
n
2.6
⎞2
uj xrj /2−1 ⎠ ≥ 0 ,
x > 0.
j1
This shows that μ is a convex function for x > 0. By setting f μ in 1.5, respectively and
from Remark 1.3, we get
n
uj uk
pφrjk a qφrjk b
pq
j,k1
n
j,k1
n
j,k1
uj uk
uj uk
1
2y
M
1
−
2y
M
φrjk xdx − φrjk
m
pq
φrjk
φrjk xdx
≥ 0,
m
pφrjk a qφrjk b
pa qb
pq
pa qb
pq
1
−
y
M
≥ 0,
2.7
φrjk xdx
≥ 0,
m
or equivalently
n
j,k1
uj uk i rjk ≥ 0.
2.8
6
Journal of Inequalities and Applications
Therefore the given matrix is a positive semidefinite. By using well-known Sylvester criterion,
we have
rj rk p
≥ 0 for each 1 ≤ p ≤ n.
det i
2
j,k1
2.9
ii Since limr → l i r i l for l 0, 1, it follows that i is continuous on Ê. Therefore,
by Proposition 1.5 for f i , we get exponential convexity of i on Ê.
iii Let i r > 0, then the log-convexity of i is a simple consequence of Corollary 1.7.
By setting f log i , s1 r, s2 s, s3 t in Lemma 1.8, we have
t − r log i s ≤ t − s log i r s − ri t,
2.10
which implies 2.4.
We will use the following lemma in the proof of mean value theorem.
Lemma 2.3 see 1, page 4. Let f ∈ C2 a, b such that
α ≤ f x ≤ β
∀x ∈ a, b.
2.11
If one considers the functions h1 , h2 , defined by
h1 x αx2
− fx,
2
2.12
βx2
h2 x fx −
,
2
then h1 and h2 are convex on a, b.
Proof. Therefore
h
1 x α − f x ≥ 0,
2.13
h
2 x f x − β ≥ 0,
that is, hj 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 ∈ C2 a, b
then there exists ξ ∈ a, b such that
f ξ i 2
F i f; p, q; a, b, y F x ; p, q; a, b, y
2
for i 1, 2, 3.
2.14
Journal of Inequalities and Applications
7
Proof. Since f ∈ C2 a, b, we can take that α ≤ f ≤ β. Now in Remark 1.3, replacing f by
hj , j 1, 2 defined in Lemma 2.3, we have
F i hj ; p, q; a, b, y ≥ 0
for j 1, 2.
2.15
This gives
β F i fx; p, q : a, b, y ≤ F i x2 ; p, q; a, b, y ,
2
α i 2
F x ; p, q; a, b, y ≤ F i fx; p, q; a, b, y .
2
2.16
Combining 2.16 and 14, we get
β α i 2
F x ; p, q; a, b, y ≤ F i fx; p, q; a, b, y ≤ F i x2 ; p, q; a, b, y .
2
2
2.17
By using Remark 1.2
F i x2 ; p, q; a, b, y > 0,
2.18
therefore
2F i fx; p, q; a, b, y
≤ β.
α≤
F i x2 ; p, q; a, b, y
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 ∈ C2 a, 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 ξ
g ξ
F i g; p, q; a, b, y
for i 1, 2, 3.
2.20
Proof. Define functions φi ∈ C2 a, b, i 1, 2, 3 by
φi c1i g − c2i f,
2.21
8
Journal of Inequalities and Applications
where
c1i F i f; p, q; a, b, y ,
c2i F i g; p, q; a, b, y .
2.22
Then using Theorem 2.4 for f φi , we have
0
g
c1i
ξ
i f ξ
− c2
F i x2 ; p, q; a, b, y .
2
2
2.23
Using Remark 1.2
F i x2 ; p, q; a, b, y > 0,
2.24
therefore
c1i
c2i
f ξ
,
g ξ
2.25
which is clearly 2.20.
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
tt − 1F i xr ; p, q; a, b, y
rr − 1F i xt ; p, q; a, b, y
for i 1, 2, 3.
2.26
Remark 2.7. If the inverse of f /g exists, then from 2.20 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
Journal of Inequalities and Applications
9
3. Means of Stolarsky Type
Expression 2.27 gives the means. We can consider
i Er,t
p, q; a, b, y
1/r−t
F i φr ; p, q; a, b, y
,
F i φt ; p, q; a, b, y
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 of 3.1:
⎧
par qbr Mr1 − mr1
⎪
1
⎪
⎪
−
,
⎪
⎪
rr − 1
pq
2yr 1
⎪
⎪
⎪
⎪
⎪
pb qa
log M − log m
⎪
⎪
,
−
⎪
⎪
⎪
4y
2ab
p
q
⎪
⎪
⎨
1 r Mlog M − 1 − mlog m − 1 p log a q log b
⎪
⎪
−
,
⎪
⎪
2y
pq
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
pa log a qb log b
⎪
⎪
− Γ,
⎪
⎪
pq
⎪
⎩
r/
− 1, 0, 1,
r −1,
r 0,
r 1,
⎧
⎪
pa qb r
Mr1 − mr1
1
⎪
⎪
,
r/
− 1, 0, 1,
−
⎪
⎪
⎪
rr − 1
2yr 1
pq
⎪
⎪
⎪
⎪
pq
log M − log m
⎪
⎪
− r −1,
,
⎨
4y
2 pa
qb
2 r ⎪
M log M − 1 − m log m − 1
pa qb
⎪
⎪
⎪log
−
, r 0,
⎪
⎪
pq
2y
⎪
⎪
⎪
⎪
pa qb
pa qb
⎪
⎪
log
,
r 1,
⎩Γ −
pq
pq
⎧
pa qb r Mr1 − mr1
par qbr
⎪
1
⎪
⎪
,
−
⎪
⎪
rr − 1
pq
pq
yr 1
⎪
⎪
⎪
⎪
⎪
pb qa
pq
log M − log m
⎪
⎪
,
−
⎪
⎪
⎪
2y
2ab p q
2 pa qb
⎪
⎪
⎨
3 r Mlog M − 1 − mlog m − 1 p log a q log b
pa qb
⎪
⎪
−
−
log
,
⎪
⎪
y
pq
pq
⎪
⎪
⎪
⎪
⎪
⎪
⎪
pa log a qb log b pa qb
pa qb
⎪
⎪
⎪
log
−
1
− 2Γ,
⎪
⎪
pq
pq
pq
⎪
⎩
r/
− 1, 0, 1,
r −1,
r 0,
r 1,
3.2
where Γ M2 2 log M − 1 − m2 2 log m − 1/8y.
10
Journal of Inequalities and Applications
We have
⎧ 1/r−t
⎪
F i φr ; p, q; a, b, y
⎪
⎪
⎪
,
⎪
⎪
i φ ; p, q; a, b, y
F
⎪
t
⎪
⎪
⎪
⎪
F i φ0 φr ; p, q; a, b, y
1 − 2r
⎪
⎪
⎪
,
⎪
⎪exp rr − 1 − F i φr ; p, q; a, b, y
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
3 F i φ0 φ−1 ; p, q; a, b, y
⎪
⎨
exp
−
,
i
2
F i φ−1 ; p, q; a, b, y
p, q; a, b, y Er,t
⎪
⎪
⎪
2
⎪
⎪
i
⎪
⎪
⎪exp 1 − F φ0 ; p, q; a, b, y ,
⎪
⎪
⎪
2F i φ0 ; p, q; a, b, y
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
F i φ0 φ1 ; p, q; a, b, y
⎪
⎪exp −1 −
,
⎪
⎪
⎪
2F i φ1 ; p, q; a, b, y
⎪
⎩
r
/ t,
r t
/ − 1, 0, 1,
r t −1,
3.3
r t 0,
r t 1,
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:
fr
ft
1/r−t
≤
fu
fv
1/u−v
.
3.4
The proof of this Lemma is given in 1.
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:
i i Er,t
p, q; a, b, y ≤ Eu,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 inequality 1, page 2:
φx1 − φx2 φ y2 − φ y1
≤
,
x2 − x1
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 log i u − log i v
≤
,
r −t
u−v
3.7
which is equivalent to 3.5 for t /
r, u /
v. By continuity of i , 3.5 is valid for t r, u v.
Journal of Inequalities and Applications
11
i
p, q; a, b, y,
Remark 3.3. If we substitute p q 1 and replace r → r − 1 and t → t − 1 in Er,t
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 → as , b → bs , y → ys , r → r/s, t → t/s, ξ → ξ1/s in 2.26, we get
ξ
r−t
tt − sF i xr/s ; p, q; as , bs , ys
,
rr − sF i xt/s ; p, q; as , bs , ys
s/
0, t /
r for i 1, 2, 3.
4.1
It follows that
i
p, q; as , bs , ys
Er,t;s
1/r−t
F i φr/s ; p, q; as , bs , ys
,
F i φt/s ; p, q; as , bs , ys
s
/ 0, t / r for i 1, 2, 3.
4.2
To get all continuous extension of 4.2, we consider
⎧
s
s 1/s
⎪
⎨ pa qb
, s
/ 0,
A
pq
⎪
⎩ap bq 1/pq ,
s 0,
⎧
⎪
1/s
bs − as
⎪
⎪
min p, q
, s/
0,
⎨
q
y ≤ p1/pq
min{p,q}
⎪
b
⎪
⎪
,
s 0.
⎩
a
4.3
For s / 0, we define
is r F i φr/s ; p, q; as , bs , ys
for i 1, 2, 3,
4.4
where {φr ; r ∈ Ê} is the family of functions defined in Lemma 2.1. Here we
have F i f; p, q; as , bs , ys defined as
pfas qfbs 1 Ms
F f; p, q; a , b , y − s
fxdx,
pq
2y ms
s
pa qbs
1 Ms
2
s s
s
,
fxdx − f
F f; p, q; a , b , y 2ys ms
pq
s
pa qbs
pfas qfbs 1 Ms
− s
F 3 f; p, q; as , bs , ys f
fxdx,
pq
pq
y ms
1
s
s
s
where i 1, 2, 3, ms As − ys , and Ms As ys .
4.5
12
Journal of Inequalities and Applications
We have
⎧
rs/s
rs/s
par qbr
s Ms
− ms
s2
⎪
⎪
⎪
,
−
⎪
⎪
rr − s
pq
r s
2ys
⎪
⎪
⎪
⎪
⎪
⎪
⎪
log Ms − log ms
⎪ pbs qas
⎪
⎪
−
,
⎪
⎨ 2as bs p q
4ys
1
s r ⎪
⎪
Ms log Ms − 1 − ms log ms − 1
p log a q log b
⎪
⎪
⎪
,
−s
⎪
s
⎪
2y
pq
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪ pas log a qbs log b
⎪
⎩s
− Γs ,
pq
2s r
⎧
s
⎪
pa qbs r/s
s Ms rs/s − ms rs/s
s2
⎪
⎪
−
,
⎪
⎪
rr − s r s
2ys
pq
⎪
⎪
⎪
⎪
⎪
s
⎪
⎪
⎪
log Ms − log ms
pq
⎪
⎪
⎪
−
,
⎪
⎨
4ys
2 pas qbs
⎪
⎪
⎪
Ms log Ms − 1 − ms log ms − 1
pas qbs
⎪
⎪
,
−
s
log
⎪
⎪
⎪
2ys
pq
⎪
⎪
⎪
⎪
⎪
s
s
⎪
⎪
pa qbs s
pa qbs
⎪
⎪
,
log
⎩Γs − s
pq
pq
r/
− s, 0, s,
r −s,
r 0,
r s,
r/
− s, 0, s,
r −s,
r 0,
r s,
⎧
s
rs/s
rs/s
⎪
s2
pa qbs r/s
par qbr
− ms
s Ms
⎪
⎪
,
−
⎪
⎪
rr − s
⎪
pq
pq
rs
ys
⎪
⎪
⎪
⎪
⎪
⎪
r
⎪
/ − s, 0, s;
⎪
⎪
⎪
⎪
⎪
⎪
s
⎪
s
s
⎪
⎪
pq
log Ms − log ms
⎪ pb qa ⎪
s
−
,
r −s,
⎪
⎪
s
⎨ 2as bs p q
2ys
2 pa qb
3
s r s
⎪ M log M − 1 − m log m − 1
⎪
pa qbs
s
s
s
s
⎪
⎪
⎪
− s log
⎪
⎪
ys
pq
⎪
⎪
⎪
⎪
⎪
⎪
p log a q log b
⎪
⎪
⎪
−s
,
r 0,
⎪
⎪
pq
⎪
⎪
⎪
⎪
s
s
⎪
⎪
pa log a qb log b
pa qbs s
pa qbs
⎪
⎪
⎩s
s
− 2Γs ,
log
r s,
pq
pq
pq
where Γs Ms 2 2 log Ms − 1 − ms 2 2 log ms − 1/8ys .
4.6
Journal of Inequalities and Applications
13
For s 0, we consider a family of convex functions {ψr : r ∈ Ê} defined on
⎧
1
⎪
⎨ 2 erx ,
r
ψr x ⎪
⎩ 1 x2 ,
2
Ê
r/
0,
by
4.7
r 0.
We have F i f; p, q; log a, log b, log y, i 1, 2, 3 defined as
log M
pf log a qf log b
1
1
−
F f; p, q; log a, log b, log y fxdx,
pq
2 log y log m
log M
p log a q log b
1
2
,
fxdx − f
F f; p, q; log a, log b, log y 2 log y log m
pq
pf log a qf log b
p log a q log b
F 3 f; p, q; log a, log b, log y f
pq
pq
log M
1
−
fxdx,
log y log m
4.8
where log m logap bq 1/pq /y, log M logyap bq 1/pq . Now for
i0 r F i ψr ; p, q; log a, log b, log y
for i 1, 2, 3,
⎧ r/pq 2r
⎪
y −1
1 par qbr ap bq ⎪
⎪
⎪
−
,
⎨ r2
pq
2ryr log y
10 r 2
2
⎪
1
1 p log a q log b
⎪
2
2
p q 1/pq
⎪
− log a b − log y ,
⎪
⎩2
pq
3
r/
0,
r 0,
⎧ p q r/pq 2r
y −1
⎪
1 a b r/pq
⎪
p
q
⎨
, r/
0,
− a b 2
2ryr log y
20 r r
⎪
⎪
⎩ 1 log2 y,
r 0,
6
⎧ ⎪
ap bq r/pq y2r − 1
1 par qbr
⎪
p q r/pq
⎪
a b ,
⎪
−
⎨ r2
pq
ryr log y
30 r 2
2
⎪
1 p log a q log b
2
⎪
⎪
− log2 ap bq 1/pq − log2 y ,
⎪
⎩2
pq
3
4.9
r/
0,
r 0.
14
Journal of Inequalities and Applications
We get means
⎧ 1/r−t
i
s s
s
⎪
⎪
⎪ F φr/s ; p, q; a , b , y ⎪
⎪
⎪
i φ ; p, q; as , bs , y s
⎪
F
t/s
⎪
⎪
⎪
⎪
F i φ0 φr/s ; p, q; as , bs , ys
⎪
r
− 2s
⎪
⎪
−
exp
⎪
⎪
rr − s
⎪
sF i φr/s ; p, q; as , bs , ys
⎪
⎪
⎪
⎪
⎪
⎪
⎪
i
s s
s
⎪
⎪
⎪exp 3 − F φ0 φ−1 ; p, q; a , b , y ⎪
⎪
⎪
2s
sF i φ−1 ; p, q; as , bs , ys
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
F i φ02 ; p, q; as , bs , ys
⎪
1
⎪
⎪
⎪
⎨exp s − 2sF i φ ; p, q; as , bs , ys 0
i
s s
s
Er,t;s p, q; a , b , y ⎪
⎪
⎪
⎪
1 F i φ0 φ1 ; p, q; as , bs , ys
⎪
⎪
⎪exp − −
⎪
⎪
s 2sF i φ1 ; p, q; as , bs , ys
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
F i xψr ; p, q; log a, log b, log y
2
⎪
⎪
⎪exp − −
⎪
⎪
r
⎪
Fi ψr ; p, q; log a, log b, log y
⎪
⎪
⎪
⎪
⎪
i
⎪
⎪
⎪
F xψ0 ; p, q; log a, log b, log y
⎪
⎪
⎪exp
⎪
⎪
3F i ψ0 ; p, q; log a, log b, log y
⎪
⎪
⎪
⎪
⎪
⎩
r/
t, s ∈ Ê \ {0},
r t, r 2 − rs /
0,
r t −s, s /
0,
r t 0, s /
0,
r t s, s /
0,
r t
/ 0, s 0,
r t s 0,
4.10
for i 1, 2, 3.
Theorem 4.1. Theorem 2.2 is still valid if one sets φr ψr .
Proof. The proof is similar to the proof of Theorem 2.2.
Theorem 4.2. Let p, q, a, b, A, and y are real numbers as defined in Theorem 1.1 also let r, t, u, v ∈ Ê
such that r ≤ u, t ≤ v, then the following inequality is valid:
i
i
Er,t;s
p, q; as , bs , ys ≤ Eu,v;s
p, q; as , bs , ys
for i 1, 2, 3.
4.11
Proof. For s /
0, in this case we use Lemma 3.1 for f i , and we have that
i r
i t
1/r−t
≤
i u
i v
1/u−v
,
4.12
Journal of Inequalities and Applications
15
t, u /
v. For s > 0, by substituting a → as , b → bs , r → r/s,
for t, r, u, v ∈ Ê, r ≤ u, t ≤ v, r /
t → t/s, u → u/s, v → v/s, such that r/s ≤ u/s, t/s ≤ v/s, t /
r, u /
v in 4.12, we get
is r
is t
s/r−t
≤
is u
is v
s/u−v
.
4.13
For s < 0, by substituting i r is r, a → as , b → bs , r → r/s, t → t/s, u → u/s,
v → v/s, such that u/s ≤ r/s, v/s ≤ t/s, in 4.12 we have
is u
is v
s/u−v
≤
is r
is t
s/r−t
,
4.14
By raising power 1/s, to 4.13 and −1/s, to 4.14, we get 4.11 for t /
r, u /
v.
For s 0, since i0 r is log-convex function, therefore Lemma 3.1 implies that for
r ≤ u, t ≤ v, t / r, u / v, we have
i
i
Er,t;0
p, q; log a, log b, log y ≤ Eu,v;0
p, q; log a, log b, log y ,
4.15
which completes the proof.
Remark 4.3. If we substitute p q 1, s → s − 1, and t → t − 1 in the above results, then the
results of generalized Stolarsky type means proved in 6 are recaptured.
Acknowledgment
This research was partially funded by Higher Education Commission, Pakistan. The research
of the rst author was supported by the Croatian Ministry of Science, Education and Sports
under the Research Grant 117-1170889-0888.
References
1 J. E. Pečarić, 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.
2 S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications,
RGMIA Monograph Collections, Victoria University, 2002.
3 P. M. Vasić and I. B. Lacković, “On an inequality for convex functions,” Univerzitet u Beogradu.
Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika, no. 461–497, pp. 63–66, 1974.
4 A. Lupaş, “A generalization of Hadamard inequalities for convex functions,” Univerzitet u Beogradu.
Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika, no. 544–576, pp. 115–121, 1976.
5 P. M. Vasić and I. B. Lacković, “Some complements to the paper: “On an inequality for convex
functions”,” Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika, no.
544–576, pp. 59–62, 1976.
6 J. Jakšetić, J. Pečarić, and A. ur Rehman, “On Stolarsky and related means,” Mathematical Inequalities
and Applications, vol. 13, pp. 899–909, 2010.
7 M. Anwar, J. Jakšetić, J. Pečarić, and A. ur Rehman, “Exponential convexity, positive semi-definite
matrices and fundamental inequalities,” Journal of Mathematical Inequalities, vol. 4, no. 2, pp. 171–189,
2010.
Download