Volume 2009, Article ID 628051,16pages doi:10.1155/2009/628051
Research Article
Cauchy Means of the Popoviciu Type
Matloob Anwar,
1Naveed Latif,
1and J. Pe ˇcari´c
1, 21Abdus Salam School of Mathematical Sciences, 68-B New Muslim Town, GC University,
Lahore 54000, Pakistan
2Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, Zagreb 10000, Croatia
Correspondence should be addressed to Naveed Latif,sincerehumtum@yahoo.com
Received 9 October 2008; Accepted 2 February 2009
Recommended by Wing-Sum Cheung
We discuss log-convexity for the differences of the Popoviciu inequalities and introduce some mean value theorems and related results. Also we give the Cauchy means of the Popoviciu type and we show that these means are monotonic.
Copyrightq2009 Matloob Anwar et al. 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 and Preliminaries
Letfxandpxbe two positive real valued functions withabpxdx1, then from theory of convex meanscf.1–3, the well-known Jensen inequality gives that fort <0 ort >1,
b
a
pxfxtdx≥
b
a
pxfxdx
t
, 1.1
and vise versa for 0< t <1. In4, Simic has considered the difference
DsDsa, b, f, p
b
a
pxfxsdx−
b
a
pxfxdx
s
. 1.2
Theorem 1.1. Let fx, px be nonnegative and integrable functions for x ∈ a, b, with
b
apxdx1, then for0< r < s < t;r, s, t /1, one has
Ds
ss−1
t−r
≤
Dr
rr−1
t−s D t
tt−1
s−r
. 1.3
Remark 1.2. For extension ofTheorem 1.1seecf.4.
Popoviciu6–8,9, pages 214-215has proved the following results.
Theorem 1.3. Letφ : a, b → Rbe convex andf : 0,1 → Rbe continuous, increasing, and convex such thata≤fx≤bforx0,1. Then
1
0
φfxdx≤ ba−2f b−a φa
2f−a b−a2
b
a
φxdx, 1.4
where
f
1
0
fxdx. 1.5
Ifφis strictly convex, then the equality in1.4holds if and only if
fx a b−ax−λ|x−λ|
21−λ , whereλ
ba−2f
b−a . 1.6
Theorem 1.4. Letφ :a, b → Rbe continuous and convex, and letf : 0,1 → Rbe convex of order1, . . . , n1such thata≤fx≤bforx0,1.
Then
1
0
φfxdx≤
1
0
φUjf, x
dx for jab j1 ≤f≤
j−1ab
j , 2≤j≤n, 1.7
1
0
φfxdx≤Vf fora≤f≤ nab
n1 , 1.8
where
Ujt, x ajj1
t− jab j1
j−1ab
j −t
x
xj−1, 1.9
Vx bna−n1x
b−a φa
n1x−a nb−an1/n
b
a
φxdx
Ifφis strictly convex, then equality in1.7holds if and only if
fx Ujf, x ; 1.11
and equality in1.8holds if
fx a b−a
x−λ|x−λ|
21−λ
n
, whereλ bna−n1f
b−a . 1.12
With the help of the following useful lemmas we prove our results.
Lemma 1.5. Define the function
ϕsx
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
xs
ss−1, s /0,1;
−logx, s0;
xlogx, s1.
1.13
Thenϕsx xs−2, that is,ϕsxis convex forx >0.
The following lemma is equivalent to definition of convex functionsee9, page 2.
Lemma 1.6. Ifφis a convex function onIfor alls1, s2, s3 ∈Ifor whichs1 < s2< s3, the following is valid
φs1
s3−s2
φs2
s1−s3
φs3
s2−s1
≥0. 1.14
We quote here another useful lemma from log-convexity theorycf.4.
Lemma 1.7. A positive function f is log-convex in the Jensen-sense on an open intervalI, that is, for eachs, t∈I,
fsft≥f2
st
2
, 1.15
if and only if the relation
u2fs 2uwf
st
2
w2ft≥0, 1.16
The following lemma given in 10gives the relation between Beta function β and Hypergeometric functionF.
Lemma 1.8. Supposea, b, c, α, γ∈Rare such thatac > b >0and0< α <2γ,βandFare Beta and Hypergeometric functions, respectively. Then
∞
0
xb−1
1αxa1γxcdxγ
−bβb, ac−bF
a, b ac
γ−γα. 1.17
The paper is organized in the following way. After this introduction, in the second section we discuss the log-convexity of differences of the Popoviciu inequalities1.4,1.7, and1.8. In the third section we introduce some mean value theorems and the Cauchy means of the Popoviciu-type and discuss its monotonicity.
2. Main Results
Theorem 2.1. Letf:0,1 → Rbe continuous, increasing, and convex such that0< a≤fx≤b forx∈0,1, and letfbe defined in1.5and
Ωsf
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
1
ss−1
ba−2f
b−a a
s 2f−a
b−a2s1
bs1−as1−
1
0
fxsdx
, s /0,1;
2f−a
b−a
1
0
logfxdx−ba−2f b−a loga−
2f−a
b−a2blogb−aloga, s0;
ba−2f
b−a aloga
f−a b−a2
b2logb−a2loga
−f−aba
2b−a −
1
0
fxlogfxdx, s1,
2.1
and letΩsfbe positive.
One has thatΩsfis log-convex and the following inequality holds for−∞< r < s < t <∞,
Ωt−r
s f≤Ωtr−sfΩst−rf. 2.2
Proof. Consider the function defined by
ωx u2ϕsx 2uwϕrx w2ϕtx, 2.3
wherer st/2,ϕsis defined by1.13andu, w∈R. We have
ωx u2xs−22uwxr−2w2xt−2
uxs/2−1wxt/2−12>0, x >0.
Therefore,ωxis convex forx >0. UsingTheorem 1.3,
ba−2f b−a
u2ϕ
sa 2uwϕra w2ϕta
2f−a b−a2
b
a
u2ϕsx 2uwϕrx w2ϕtx
dx
≥ 1
0
u2ϕsfx 2uwϕrfx w2ϕtfx
dx,
u2
ba−2f b−a ϕsa
2f−a b−a2
b
a
ϕsxdx−
1
0
ϕsfxdx
2uw
ba−2f b−a ϕra
2f−a b−a2
b
a
ϕrxdx−
1
0
ϕrfxdx
w2
ba−2f b−a ϕta
2f−a b−a2
b
a
ϕtxdx−
1
0
ϕtfxdx
≥0,
2.5
since
Ωsf b
a−2f b−a ϕsa
2f−a b−a2
b
a
ϕsxdx−
1
0
ϕsfxdx, f
1
0
fxdx, 2.6
we have
u2Ωsf 2uwΩrf w2Ωtf≥0. 2.7
ByLemma 1.7, we have
ΩsfΩtf≥Ω2rf Ω2st/2f, 2.8
that isΩsfis log-convex in the Jensen-sense fors∈R. Since
lim
s→0Ωsf Ω0f, slim→1Ωsf Ω1f. 2.9
This impliesΩsfis continuous, therefore it is log-convex.
Since Ωsf is log-convex, that is, logΩsf is convex, therefore by Lemma 1.6 for
−∞< r < s < t <∞and takingφslogΩs, we get
logΩts−rf≤logΩrt−sf logΩts−rf, 2.10
Theorem 2.2. Letf,Ωsfbe defined inTheorem 2.1and lett, s, u, vbe real numbers such that
s≤u,t≤v,s /t,u /v, one has
Ω
tf
Ωsf
1/t−s
≤ Ω
vf
Ωuf
1/v−u
. 2.11
Proof. Incf.9, page 2, we have the following result for convex functionf withx1 ≤ y1, x2≤y2,x1/x2,y1/y2:
fx2
−fx1
x2−x1 ≤ fy2
−fy1
y2−y1 .
2.12
Since byTheorem 2.1,Ωsfis log-convex, we can set in2.12:
fx logΩxandx1s,x2t,y1u,y2v. We get
logΩtf−logΩsf
t−s ≤
logΩvf−logΩuf
v−u ,
log
Ω
tf
Ωsf
1/t−s
≤log
Ω
vf
Ωuf
1/v−u
,
2.13
and after applying exponential function, we get2.11.
Theorem 2.3. Letf :0,1 → Rbe convex of order1, . . . , n1such that0 < a ≤fx≤ bfor x∈0,1, and letfbe defined in1.5and
Λsf
1
0 ϕs
Uj f, x
dx−
1
0
ϕsfxdx, 2.14
where
Uj f, x
ajj1
f−jab j1
j−1ab
j −f
x
xj−1, 2.15
for
jab j1 ≤f≤
j−1ab
j , 2≤j≤n, 2.16
and letΛsfbe positive.
One has thatΛsfis log-convex and the following inequality holds for−∞< r < s < t <∞,
Λt−r
s f≤Λtr−sfΛst−rf. 2.17
Theorem 2.4. Letf,Λsfbe defined inTheorem 2.3andt, s, u, vbe real numbers such thats≤u,
t≤v,s /t,u /v, one has
Λ
tf
Λsf
1/t−s
≤ Λ
vf
Λuf
1/v−u
. 2.18
Proof. Similar to the proof ofTheorem 2.2.
Lemma 2.5. Letf :0,1 → Rbe convex of order1, . . . , n1such that0 < a≤fx≤bforx∈ 0,1,fbe defined in1.5andV be defined in1.10, and letβandFare Beta and Hypergeometric functions respectively, and
Γsf V f
−
1
0
ϕsfxdx. 2.19
Then
Γsf
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1
ss−1
bna−n1f
b−a a
s n1f−a
nb−an1/na
s1/n
b−a b
1/n
×β
1
n,1
F ⎛ ⎜ ⎜ ⎝
s1 n1,
1
n
1
n1
b−a b ⎞ ⎟ ⎟ ⎠− 1 0
ftsdt
⎤ ⎥ ⎥
⎦, s /0,1;
bna−n1f
b−a −loga−
n1f−a nb−an1/n
× ⎡ ⎢
⎣nlogbb−a1/n−na1/n
b−a b
1/n1 β
1
n1,1
F ⎛ ⎜ ⎝ 1
n1,
1
n1
1
n2
b−a b ⎞ ⎟ ⎠ ⎤ ⎥ ⎦ 1 0
logftdt, s0;
bna−n1f
b−a aloga
n1f−a nb−an1/n
×
nab−a1/nlogb n
n1logbb−a
1/n1−a1/n
b−a b
1/n1 β
1
n1,1
×F ⎛ ⎜ ⎝ 1
n1,
1
n1
1
n2
b−a b ⎞ ⎟ ⎠ n n1 na
⎤ ⎥ ⎦−
1
0
ftlogftdt, s1,
for
a≤f≤ nab
n1 . 2.21
Proof. First, we solve these three integrals
I1
b
a
ts
t−an−1/ndt, I2
b
a
logt
t−an−1/ndt, I3
b
a
tlogt
t−an−1/ndt. 2.22
Take
I1
b
a
ts
t−an−1/ndt. 2.23
Substitute
t abx
1x , dt b−a
1x2dx, A
and limits, whent → athenx → 0, whent → bthenx → ∞. So,
I1
b
a
ts
t−an−1/ndt
∞
0
abx/1xs abx/1x−a1−1/n·
b−a 1x2dx
b−a1/n
∞
0
abxs1x1−1/n 1xs2x1−1/n dx,
I1 b−a1/nas
∞
0
x1/n−1
1xs1/n11 b/ax−sdx.
2.24
By usingLemma 1.8withas1/n1,b1/n,c−s,α1,γb/asuch that 1/n1>
1/n >0 and 0<1<2b/a, we get
I1 as1/nb−a1/nβ
1
n,1
F
⎛ ⎜ ⎜ ⎝
s 1 n1,
1
n
1
n1
b−a b
⎞ ⎟ ⎟
⎠. 2.25
Take second integral
I2
b
a
logt
using integration by parts, we have
I2nlogbb−a1/n−n
b
a
t−1t−a1/ndt. 2.27
Let
I4
b
a
t−1t−a1/ndt. 2.28
By using same substitutionAas above, we get
I4
∞
0
1x abx ·
abx
1x −a
1/n
· b−a
1x2dx
b−a1/n1 a
∞
0
x1/n1−1
1x1/n11 b/axdx. 2.29
By usingLemma 1.8witha1/n1,b1/n1,c1,α1,γb/asuch that 1/n2 >
1/n1>0 and 0<1<2b/a, we get
I2nlogbb−a1/n−na1/n
b−a b
1/n1 β
1
n1,1
F
⎛ ⎜ ⎜ ⎝
1
n1,
1
n1
1
n2
b−a b
⎞ ⎟ ⎟
⎠. 2.30
Now, take third integral
I3
b
a
tlogt
t−an−1/ndt. 2.31
Using integration by parts, we get
I3nab−a1/nlogb n
n1logbb−a
1/n1− n n1
b
a
t−a1/n1
t dt−na
b
a
t−a1/n
t dt.
2.32
Let
I5
b
a
t−a1/n1
t dt. 2.33
By using same substitutionAas above, we get
I5
∞
0
abx/1x−a1/n1 abx/1x ·
b−a 1x2dx
b−a1/n1 a
∞
0
x1/n1−1
By usingLemma 1.8witha1/n1,b1/n1,c1,α1,γb/asuch that 1/n2 >
1/n1>0 and 0<1<2b/a, we get
I5a1/n
b−a b
1/n1 β
1
n1,1
F
⎛ ⎜ ⎜ ⎝
1
n1,
1
n1
1
n2
b−a b
⎞ ⎟ ⎟
⎠. 2.35
Let
I6
b
a
t−a1/n
t dt. 2.36
By using same substitutionAas above, we have
I6
∞
0
abx/1x−a1/n abx/1x ·
b−a 1x2dx
b−a1/n1 a
∞
0
x1/n1−1
1x1/n11 b/axdx. 2.37
By usingLemma 1.8witha1/n1,b1/n1,c1,α1,γb/asuch that 1/n2 >
1/n1>0 and 0<1<2b/a, we get
I6a1/n
b−a a
1/n1 β
1
n1,1
F
⎛ ⎜ ⎜ ⎝
1
n1,
1
n1
1
n2
b−a b
⎞ ⎟ ⎟
⎠. 2.38
Then
I3nab−a1/nlogb n
n1logbb−a
1/n1− a1/n
b−a a
1/n1
×β
1
n1,1
F
1
n1,
1
n1,
1
n2
b−bann1 na
.
2.39
Fors /0,1,
Γsf
1
ss−1
bna−n1f
b−a a
s n1f−a
nb−an1/n
b
a
ts
t−an−1/ndt−
1
0
ftsdt
.
UsingI1, we have
Γsf 1
ss−1
bna−n1f
b−a a
s n1f−a
nb−an1/na
s1/n
b−a b
1/n
×β
1
n,1
F ⎛ ⎜ ⎜ ⎝
s1 n1,
1
n
1
n1
b−a b ⎞ ⎟ ⎟ ⎠− 1 0
ftsdt
⎤ ⎥ ⎥ ⎦.
2.41
Fors0,
Γ0f
bna−n1f
b−a −loga−
n1f−a nb−an1/n
b
a
logt t−an−1/ndt
1
0
logftdt. 2.42
UsingI2, we have
Γ0f bna−n
1f
b−a −loga−
n1f−a nb−an1/n
× ⎡ ⎢ ⎢
⎣nlogbb−a1/n−na1/n
b−a b
1/n1 β
1
n1,1
F ⎛ ⎜ ⎜ ⎝ 1
n1,
1
n1
1
n2
b−a b ⎞ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎦ 1 0
logftdt.
2.43
Fors1,
Γ1f
bna−n1f
b−a aloga
n1f−a nb−an1/n
b
a
tlogt t−an−1/ndt−
1
0
ftlogftdt.
2.44
UsingI3, we have
Γ1f
bna−n1f
b−a aloga
n1f−a nb−an1/n
×
nab−a1/nlogb n
n1logbb−a
1/n1− a1/n
b−a b
1/n1 β
1
n1,1
×F ⎛ ⎜ ⎜ ⎝ 1
n1,
1
n1
1
n2
b−a b ⎞ ⎟ ⎟ ⎠ n n1na
⎤⎥ ⎥ ⎦−
1
0
ftlogftdt.
Theorem 2.6. Letf :0,1 → Rbe convex of order1, . . . , n1such that0 < a ≤fx≤ bfor x∈0,1,fbe defined in1.5and letΓsfdefined in2.20be positive.
One has thatΓsfis log-convex and the following inequality holds for−∞< r < s < t <∞,
Γt−r
s f≤Γtr−sfΓst−rf. 2.46
Proof. As in the proof ofTheorem 2.1, we useTheorem 1.4instead ofTheorem 1.3.
Theorem 2.7. Letf,Γsfbe defined inTheorem 2.6andt, s, u, vbe real numbers such thats≤u,
t≤v,s /t,u /v, one has
Γ
tf
Γsf
1/t−s
≤ Γ
vf
Γuf
1/v−u
. 2.47
Proof. Similar to the proof ofTheorem 2.2.
3. Cauchy Means
Let us note that 2.11 has the form of some known inequalities between means e.g., Stolarsky’s means, etc.. Here we prove that expressions on both sides of2.11 are also means.
Lemma 3.1. Leth∈ C2Ibe such thath is bounded, that is,m ≤ h ≤ M.Then the functions φ1, φ2defined by
φ1t M2 t2−ht, φ2t ht−m2t2, 3.1
are convex functions.
Theorem 3.2. Leth∈C2I
1, I1 is a compact interval inRandf be a continuous, increasing and convex such thata≤fx≤bforx∈0,1,fbe defined in1.5then∃ξ∈a, b, ξ /0such that
ba−2f b−a ha
2 f−a b−a2
b
a
hxdx−
1
0
hfxdx
hξ
2
ba−2f
b−a a
22 f−a
b2baa2
3b−a −
1
0
fx2dx
.
3.2
Proof. Supposemminhx≤hx≤Mmaxhxforx∈I1. Then by applyingφ1and φ2defined inLemma 3.1forφin1.4, we have
ba−2f b−a φ1a
2 f−a b−a2
b
a
φ1xdx≥
1
0
φ1fxdx,
ba−2f b−a φ2a
2 f−a b−a2
b
a
φ2xdx≥
1
0
φ2fxdx,
that is,
M
2
ba−2f
b−a a
22 f−a
b2baa2
3b−a −
1
0
fx2dx
≥ ba−2f
b−a ha
2 f−a b−a2
b
a
hxdx−
1
0
hfxdx,
3.4
ba−2f b−a ha
2 f−a b−a2
b
a
hxdx−
1
0
hfxdx
≥ m
2
ba−2f
b−a a
22 f−a
b2baa2
3b−a −
1
0
fx2dx
.
3.5
By combining3.4and3.5and using the fact that form≤ ρ≤ Mthere existsξ ∈ I1such
thathξ ρwe get3.2.
Theorem 3.3. Letk, l∈C2I
1and satisfy3.2, f be a continuous, increasing and convex such that a≤fx≤bforx∈0,1,fbe defined in1.5, and
fx/a b−ax−λ|x−λ|
21−λ , whereλ
ba−2f
b−a , 3.6
then there existsξ∈I1such that
kξ lξ
ba−2f/b−aka 2f−a/b−a2abkxdx−10kfxdx
ba−2f/b−ala 2f−a/b−a2ablxdx−10lfxdx . 3.7
Provided that denominators are non-zero.
Proof. Consider the linear functionalsΨandηsuch thatΨm ηmξfor some function
m∈C2I
1andξ∈I1. Consider the following linear combination
mc1k−c2l, 3.8
wherec1andc2are defined as follows:
c1 Ψl ba−
2f b−a la
2 f−a b−a2
b
a
lxdx−
1
0
lfxdx,
c2 Ψk
ba−2f b−a ka
2 f−a b−a2
b
a
kxdx−
1
0
kfxdx.
3.9
Sincek, l∈C2I
1and satisfy3.2, thereforemas linear combination ofkandlshould also
Letηbe defined as follows:
ηkξ k
ξ
2
ba−2f
b−a a
22 f−a
b2baa2
3b−a −
1
0
fx2dx
. 3.10
Obviously, we have
Ψm 0. 3.11
On the other hand, there is anξ∈I1such that
ηmξ Ψm 0. 3.12
By using the linearity property of the operatorη
0 Ψlηkξ−Ψkηlξ. 3.13
NowΨl/0 andηlξ/0, we have from the last equation
Ψk Ψl
ηkξ
ηlξ. 3.14
After putting values, we get3.7.
Corollary 3.4. Let f be a continuous, increasing and convex such thata≤fx≤ bforx∈0,1, then for−∞< s /t /0, 1/s <∞there existsξ∈I1such that
ξt−s ss−1 tt−1
DatLbt1−at1/t1−1 0fx
t
dx
DasLbs1−as1/s1−1 0fx
sdx
, 3.15
whereDdenotesba−2f/b−aandLdenotes2f−a/b−a2.
Proof. Setkx ϕtxandlx ϕsx, t /s /0,1 in3.7we get3.15.
Remark 3.5. Since the functionξ →ξt−sis invertible, therefore from3.15we have
a≤
ss−1
tt−1 ·
DatLbt1−at1/t1−1 0fx
t
dx
DasLbs1−as1/s1−1 0fx
s
dx
1/t−s
In fact, similar result can also be given for3.7. Namely, suppose thatk/lhas inverse function. Then from3.7we have
ξ
k l
−1Dka Lb
akxdx−
1
0kfxdx
Dla Lablxdx−10lfxdx
. 3.17
The expression on the right-hand side of3.17is also a mean.
From the inequality3.16, we can define meansMt,sfas follows:
Mt,sf
ss−1
tt−1 ·
DatLbt1−at1/t1−1 0fx
tdx
DasLbs1−as1/s1−1 0fx
s dx
1/t−s
, 3.18
for−∞< s /t /0, 1/s <∞. Moreover we can extend these means in other cases. By limit we have
Ms,sf exp
μaslogaν/s1bs1logb−as1loga−ν/s12bs1−as1−E
μas ν/s1bs1−as1−1 0fx
s
dx
− 2s−1
ss−1
, s /0,1;
M0,0f exp
1
0logfx 2
dx210logfxdx− H −νblogb2−aloga2−G
2νb−a 210logfxdx−2μloga−2νblogb−aloga
;
M1,1f exp
K
ν/2b2logb2−a2loga2−3/2νb2logb−a2loga−H
2μalogaνb2logb−a2loga−ν/2b2−a2−21
0fxlogfxdx
3/4ν
b2−a2−2μaloga21
0fxlogfxdx
2μalogaνb2logb−a2loga−ν/2b2−a2−21
0fxlogfxdx
;
M1,0f
μaloga ν/2b2logb−a2loga−ν/4b2−a2−1
0fxlogfxdx νb−a 10logfxdx−μloga−νblogb−aloga
;
Ms,0f
μas ν/s1bs1−as1−1 0fx
s
dx
ss−1νb−a 10logfxdx−μloga−νblogb−aloga
1/s
, s /0,1;
Ms,1f
μas ν/s1bs1−as1−1 0fx
s
dx
ss−1μalogaν/2b2logb−a2loga−ν/4b2−a2−O
1/s−1
, s /0,1,
whereEdenotes10fxslogfxdx,Hdenotesμloga2,Gdenotes 2μloga,Kdenotes
μaloga2,Hdenotes10fxlogfx2dx, andOdenotes10fxlogfxdx, where
μ ba−2f
b−a , ν
2f−a
b−a2. 3.20
In our next result we prove that this new mean is monotonic.
Theorem 3.6. Lett≤u, r≤s, then the following inequality is valid
Mt,rf;w≤Mu,sf;w. 3.21
Proof. SinceΩsfis log-convex, therefore by2.11we get3.21.
Remark 3.7. Similar results of the Cauchy means and related results can also proved for Theorems2.3and2.6.
Acknowledgments
This research work is funded by Higher Education Commission Pakistan. The research of the third author is supported by the Croatian Ministry of Science, Education and Sports under the Research Grants 117-1170889-0888.
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