Volume 2008, Article ID 149286, 12 pagesdoi:10.1155/2008/149286 Research Article On the Monotonicity and Log-Convexity of a Four-Parameter Homogeneous Mean Zhen-Hang Yang Electric Grid P
Trang 1Volume 2008, Article ID 149286, 12 pages
doi:10.1155/2008/149286
Research Article
On the Monotonicity and Log-Convexity of
a Four-Parameter Homogeneous Mean
Zhen-Hang Yang
Electric Grid Planning and Research Center, Zhejiang Electric Power Test and Research Institute, Hangzhou 310014, China
Correspondence should be addressed to Zhen-Hang Yang,yzhkm@163.com
Received 13 April 2008; Accepted 29 July 2008
Recommended by Sever Dragomir
A four-parameter homogeneous mean Fp, q; r, s; a, b is defined by another approach The
criterion of its monotonicity and logarithmically convexity is presented, and three refined chains
of inequalities for two-parameter mean values are deduced which contain many new and classical inequalities for means
Copyrightq 2008 Zhen-Hang Yang 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
The so-called two-parameter mean or extended mean between two unequal positive numbers
x and y was defined first by Stolarsky 1 as
Er, s; x, y
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
sx r − y r
rx s − y s
1/r−s
x r − y r
rln x − ln y
1/r
x s − y s
sln x − ln y
1/s
exp
rlnx − y rlny
x r − y r − 1r
, r s / 0,
1.1
Trang 2It contains many mean values, for instance,
E1, 0; x, y Lx, y
⎧
⎨
⎩
x − y
lnx − ln y , x / y,
E1, 1; x, y Ix, y
⎧
⎪
⎪e
−1 x x
y y
1/x−y
, x / y,
1.3
E2, 1; x, y Ax, y x y
2,1
2;x, y
hx, y x √xy y
The monotonicity of Er, s; x, y has been researched by Stolarsky 1, Leach and Sholander2, and others also in 3 5 using different ideas and simpler methods
Qi studied the log-convexity of the extended mean with respect to parameters in6, and pointed out that the two-parameter mean is a log-concave function with respect to either parameterr or s on interval 0, ∞ and is a log-convex function on interval −∞, 0.
In7, Witkowski considered more general means defined by
Ru, v; r, s; x, y
Eu, v; x r , y r
Eu, v; x s , y s
1/r−s
1.6
further and investigated the monotonicity ofR
DenoteR : 0, ∞ and let fx, y be defined on Ω If for arbitrary t ∈ R with
tx, ty ∈ Ω, the following equation:
is always true, then the function fx, y is called an n-order homogeneous functions It
has many well properties8 10 Based on the conception and properties of homogeneous function, the extended mean was generalized to two-parameter homogeneous functions in
9, which is defined as follows
Definition 1.1 Assume f : U R × R → R is an n-order homogeneous function for
variablesx and y, continuous and first partial derivatives exist, a, b ∈ R× Rwitha / b,
p, q ∈ R × R.
If1, 1 /∈ U, then define that
Hf p, q; a, b
fa p , b p
fa q , b q
1/p−q
p / q, pq / 0,
Hf p, p; a, b lim
q→pHf a, b; p, q G f,p p q / 0,
1.8
where
G f,p G1/p
f
a p , b p
, G f x, y exp x x, y ln x yf fx, y y x, y ln y
f x x, y and f y x, y denote partial derivatives with respect to first and second variable of fx, y, respectively.
Trang 3If1, 1 ∈ U, then define further
Hf p, 0; a, b
fa p , b p
f1, 1
1/p
p / 0, q 0,
Hf 0, q; a, b
fa q , b q
f1, 1
1/q
p 0, q / 0,
Hf 0, 0; a, b lim
p→0Hf a, b; p, 0 a fx 1,1/f1,1 b fy 1,1/f1,1 p q 0.
1.10
Let fx, y Lx, y We can get two-parameter logarithmic mean, which is just
extended mean Ep, q; a, b defined by 1.1 In what follows we adopt our notations and denote byHL p, q; a, b or H L p, q or H L
Concerning the monotonicity and log-convexity of the two-parameter homogeneous functions, there are the following results
Theorem 1.2 see 9 Let fx, y be a positive n-order homogenous function defined on U
R× R and be second differentiable If I ln f xy < >0, then H f p, q is strictly increasing (decreasing) in either p or q on −∞, 0 and 0, ∞.
Theorem 1.3 see 10 Let fx, y be a positive n-order homogenous function defined on U
R× R and be third-order differentiable If
J x − yxI x < >0, where I ln f xy , 1.11
thenHf p, q is strictly convex (concave) with respect to either p or q on 0, ∞ and log-concave (log-convex) on −∞, 0.
By the above theorems we have the following
Corollary 1.4 see 10 The conditions are the same as Theorem 1.3 If1.11 holds, then H f p, 1− p is strictly decreasing (increasing) in p on 0, 1/2 and increasing (decreasing) on 1/2, 1.
If fx, y is symmetric with respect to x and y further, then the above monotone interval can be extended from 0, 1/2 to −∞, 0 and 0, 1/2, and from 1/2, 1 to 1/2, 1 and 1, ∞, respectively.
Corollary 1.5 see 10 The conditions are the same as Theorem 1.3 If 1.11 holds, then for p, q ∈
0, ∞ with p / q, the following inequalities:
G f, pq/2 < >H f p, q < > G f,p G f,q 1.12
hold For p, q ∈ −∞, 0 with p / q, inequalities 1.12 are reversed.
If fx, y is defined on R × R and symmetric with respect to x and y further, then substituting p q > 0 for p, q ∈ 0, ∞ and p q < 0 for p, q ∈ −∞, 0, 1.12 are also true, respectively.
Let fx, y Lx, y, Ax, y, Ix, y, and Dx, y in Theorems 1.2 and 1.3, Corollaries 1.4and 1.5, we can deduce some useful conclusions see 9, 10 These show the monotonicity and log-convexity ofLx, y, Ax, y, Ix, y, and Dx, y depend on the
Trang 4signs ofI ln f xyandJ x−yxI x, respectively NotingHL r, s; x, y contains Lx, y, Ax, y, and Ix, y, naturally, we could make conjecture on the similar conclusion is also true
forHf p, q; a, b, where fx, y H L r, s; x, y Namely, the monotonicity and log-convexity
of the functionHHL also depend on the signs ofI ln f xy < 0 and J x − yxI x > 0,
respectively, which is just purpose of this paper
2 Definition and main results
For stating the main results of this paper, let us introduce first the four-parameter mean as follows
Definition 2.1 Assume a, b ∈ R×Rwitha / b, p, q, r, s ∈ R×R, then the four-parameter
homogeneous mean denoted byFp, q; r, s; a, b is defined as follows:
Fp, q; r, s; a, b
La pr , b pr
La ps , b psL
a qs , b qs
La qr , b qr
1/p−qr−s
, if pqrsp − qr − s / 0, 2.1 or
Fp, q; r, s; a, b
a pr − b pr
a ps − b ps
a qs − b qs
a qr − b qr
1/p−qr−s
, if pqrsp − qr − s / 0; 2.2
ifpqrsp − qr − s 0, then the Fp, q; r, s; a, b are defined as their corresponding limits, for
example,
Fp, p; r, s; a, b lim
q→p Fp, q; r, s; a, b
Ia pr , b pr
Ia ps , b ps
1/pr−s
, if prsr − s / 0, p q;
Fp, 0; r, s; a, b lim
q→0 Fp, q; r, s; a, b
La pr , b pr
La ps , b ps
1/pr−s
, if prsr − s / 0, q 0;
F0, 0; r, s; a, b lim
p→0 Fp, 0; r, s; a, b Ga, b, if rsr − s / 0, p q 0,
2.3 whereLx, y, Ix, y are defined by 1.2, 1.3 respectively, Ga, b √ab.
It is easy to verify thatFp, q; r, s; a, b are symmetric with respect to a and b, p and
q, r and s, p, q and r, s, and then Fp, q; r, s; a, b is also denoted by Fp, q or Fr, s or Fp, q; r, s or Fa, b.
The four-parameter homogeneous meanFp, q; r, s; a, b contains many two-parameter
means mentioned in9, for example, seeTable 1
InTable 1,F2, 1; r, s; a, b is just the Gini mean is also called two-parameter arithmetic
mean, F1, 0; r, s; a, b is just the two-parameter mean or extended mean or Stolarsky mean
is also called two-parameter logarithmic mean, F1, 1; r, s; a, b is just the two-parameter
exponential mean, andF3/2, 1/2; r, s; a, b is just the two-parameter Heron mean.
Our main results can be stated as follows
Theorem 2.2 If r s > <0, then Fp, q; r, s; a, b are strictly increasing (decreasing) in either p or
q on −∞, ∞.
Trang 5Table 1: Some familiar two-parameter mean values.
p, q Fp, q; r, s; a, b p, q Fp, q; r, s; a, b
2, 1 a a r s b b r s
2,1
2
a r/2 , b r/2
Ia s/2 , b s/2
2/r−s
1, 1
Ia r , b r
Ia s , b s
1/r−s
2
3,1
3
a s/3 b s/3
3/r−s
1,1
2
a r/2 b r/2
a s/2 b s/2
2/r−s
3
4,1
4
a r/2√abr/2 b r/2
a s/2√abs/2 b s/2
2/r−s
1, 0 s r a a r s − b − b r s 1/r−s 4
3, −1
3
a r/3 b r/3
a s/3 b s/3
a2r/3 b2r/3
a2s/3 b2s/3
3/5r−s
G2/5
1, −12
a r/2 b r/2
a s/2 b s/2
2/3r−s
2, −12
a r√
abr b r
a s√abs b s
1/2r−s
√
ab1/2
3
2,1
2
a r√abr b r
a s√abs b s
1/r−s
2, −1 a a r s b b r s 1/3r−s√ab2/3
Theorem 2.3 If r s > <0, then Fp, q; r, s; a, b are strictly log-concave (log-convex) in either p
or q on 0, ∞ and log-convex (log-concave) on −∞, 0.
ByCorollary 1.4, we getCorollary 2.4
Corollary 2.4 If r s > <0, then Fp, 1 − p; r, s; a, b are strictly increasing (decreasing) in p on
−∞, 1/2 and decreasing (increasing) on 1/2, ∞.
Notice for fx, y H L r, s; x, y,
G f x, y exp x x, y ln x yf fx, y y x, y ln y
exp r − s1 x r rx − y r r −x s sx − y s s
lnx r − s1 −x r ry − y r r x s sy − y s s
lny
exp1/r−s x r
x r − y r lnx r−x r y − y r r lny r
− x s x − y s slnx s−x s y − y s s lny s
Ix r , y r
Ix s , y s
1/r−s
,
2.4
by Corollary 1.5 , we get Corollary 2.5
Trang 6Corollary 2.5 Let p / q If p qr s < 0, then
GHL, pq/2 < Fp, q; r, s; a, b < GHL,p GHL,q , 2.5
where GHL,t G1/t
HL a t , b t , GHL x, y Ix r , y r /Ix s , y s1/r−s , Ix, y is defined by 1.3.
Inequalities2.5 are reversed if p qr s > 0.
3 Lemmas
To prove our main results, we need the following three lemmas
Lemma 3.1 Suppose x, y > 0 with x / y, define
Ut :
⎧
⎪
⎪x
t y t t − y t
tx − y
−2
, t / 0,
3.1
then one has
1 U−t Ut;
2 Ut is strictly increasing in −∞, 0 and decreasing in 0, ∞.
Proof. 1 A simple computation results in part 1 of the lemma, of which details are omitted
2 By directly calculations, we get
Ut
Ut ln x ln y −
2
x tlnx − y tlny
x t − y t 2
t
2t ln
x t y t− tlnx − y tlny
x t − y t − 1
2
t
lnGx t , y t
− ln Ix t , y t
.
3.2
By the well-known inequalityIa, b >√ab, we can get part two of the lemma immediately.
The following lemma is a well-known inequality proved by Carlsonsee 11, which will be used in proof ofLemma 3.3
Lemma 3.2 For positive numbers a and b with a / b, the following inequality holds:
La, b < A 2G
√
ab b
Lemma 3.3 Suppose x, y > 0 with x / y, define
V t :
⎧
⎪
⎪x
t y t x t y t
2
t − y t
tx − y
−3
, t / 0;
3.4
Trang 7then one has
1 V −t V t;
2 V t is strictly increasing in −∞, 0 and decreasing in 0, ∞.
Proof. 1 A simple computation results in part one, of which details are omitted
2 By direct calculations, we get
Vt
V t ln x ln y
x tlnx y tlny
x t y t − 3
x tlnx − y tlny
x t − y t 3t
1x t x y t t−x t3− y x t t
lnx 1 x t y y t tx t3− y y t t
lny 3t
−x2t 4x t y t y2t
x2t − y2t lnx x2t 4x t y t y2t
x2t − y2t lny 3t
3
t −
x2t 4x t y t y2t
x2t − y2t ln x − ln y
3t2tln x − ln y x2t − y2t x2t − y2t
2tln x − ln y−
x2t 4x t y t y2t
6
.
3.5
Substitutinga, b for x2t , y2tin the above last one expression, then
Vt
V t
3
t L−1a, b La, b −
a 4√ab b
6
in whichLa, b − a 4√ab b/6 < 0 by Lemma 3.2, andL−1a, b > 0 Consequently,
Vt > 0 if t < 0 and Vt < 0 if t > 0.
The proof is completed
4 Proofs of main results
To prove our main results, it is enough to make certain the signs ofI ln HLxyandJ x− yxI x becauseFa, b; p, q; r, s HHL a, b; p, q, where H L HL r, s; x, y Er, s; x, y is
defined by1.1
Proof of Theorem 2.2 Let us observe that
Through straightforward computations, we have
I lnHLxy
xyr − s1
r2x r y r
x r − y r2 − s
2x s y s
x s − y s2
xyr − s1
r2x r y r
x r − y r2 − s
2x s y s
x s − y s2
xyx − y2
Ur − Us
4.2
Trang 8ByLemma 3.1,
Ur − Us
U|r|− U|s|
|r| − |s|
r s
which shows thatI < 0 if r s > 0 and I > 0 if r s < 0.
ByTheorem 1.2, this proof is completed
Proof of Theorem 2.3 Let us consider that
J x − yxI x
xyr − s x − y
−r3x r y r
x r y r
x r − y r3 s
3x s y s
x s y s
x s − y s3
xyx − y2
V r − V s
4.4
ByLemma 3.3,
V r − V s
V|r|− V|s|
|r| − |s|
r s
it follows thatJ > 0 if r s > 0 and J < 0 if r s < 0.
UsingTheorem 1.3, this completes the proof
Proof of Corollary 2.4 By the proof ofTheorem 2.3, there must beJ < 0 if r s < 0 Note fx, y H L r, s; x, y is symmetric with respect to x and y, it follows fromCorollary 1.4 thatFp, 1 − p; r, s; a, b HHL a, b; p, 1 − p is strictly decreasing in p on −∞, 0 and 0, 1/2.
Because
F0, 1; r, s; a, b lim
p → 0 Fp, 1 − p; r, s; a, b
La r , b r
La s , b s
1/r−s
r
a r − b r
a s − b s
1/r−s
,
4.6
thusFp, 1 − p; r, s; a, b is strictly decreasing in p on −∞, 1/2.
Likewise,Fp, 1 − p; r, s; a, b is strictly increasing in p on 1/2, ∞ if r s > 0.
This proof is completed
Proof of Corollary 2.5 By the proof of Theorem 2.3, there must J < 0 if r s < 0 Notice fx, y H L r, s; x, y is defined on R×Rand symmetric with respect tox and y, it follows
fromCorollary 1.5that2.5 holds for p q > 0 In this way, for r s < 0 and p q > 0 that
2.5 are also hold byCorollary 1.5 Hence, that2.5 are always hold for p qr s < 0.
Likewise,2.5 are reversed for p qr s > 0.
The proof ends
Trang 95 Chains of inequalities for two-parameter means
Leta and b be positive numbers The p-order power mean, Heron mean, logarithmic mean,
exponentialidentic mean, power-exponential mean, and exponential-geometric mean are defined as
M p:
⎧
⎨
⎩
M1/p
a p , b p
ifp / 0,
where L La, b, I Ia, b, A Aa, b, and h ha, b are defined by 1.2–1.5, respectively; while the power-exponential mean and exponential-geometric mean are defined
byZ : a a/ab b b/abandY : E exp1 − G2/L2, in which G Ga, b √ab, respectively
see 9, Examples 2.2 and 2.3
Concerning the above means there are many useful and interesting results, such as
L < A1/3see 12; I > A2/3see 13; Z ≥ A2see 5; h ≤ I see 14; L2 ≤ A2/3 ≤ I see
15; La, b ≤ h p a, b ≤ A q a, b hold for p ≥ 1/2, q ≥ 2p/3 see 16
Recently, Neuman applied the comparison theorem to obtain the following result Let
p, q, r, s, t ∈ R Then, the inequalities
hold true if and only ifp ≤ 2r ≤ 3s ≤ 2t see 17
It is worth mentioning that the author obtained the following chains of inequalities
see 9,10 by applying the monotonicity and log-convexity of two-parameter homogenous functions:
L2< h < A2/3 < I < Z1/3 < Y1/2 5.5 Using our main results in this paper, the above chains of inequalities can be generalized in form of inequalities for two-parameter means, which contain many classical inequalities
Example 5.1 ByTheorem 2.2, forr s > 0, we have
F1, −1; r, s; a, b < F 1, −1
2;r, s; a, b
< F1, 0; r, s; a, b
< F 1,1
2;r, s; a, b
< F1, 1; r, s; a, b < F1, 2; r, s; a, b,
5.6
that is,
G < a r/2 b r/2
a s/2 b s/2
2/3r−s
G2/3 < s
r
a r − b r
a s − b s
1/r−s
< a r/2 b r/2
a s/2 b s/2
2/r−s
<
Ia r , b r
Ia s , b s
1/r−s
< a a r s b b r s
1/r−s
,
5.7
Trang 10which can be concisely denoted by
G <
Aa r/2 , b r/2
Aa s/2 , b s/2
2/3r−s
G2/3 <
La r , b r
La s , b s
1/r−s
<
Aa r/2 , b r/2
Aa s/2 , b s/2
2/r−s
<
Ia r , b r
Ia s , b s
1/r−s
<
Aa r , b r
Aa s , b s
1/r−s
,
5.8
whereL, I, A are defined by 1.2–1.4
In particular, puttingr 1, s 0; r 2s 2; r s 1 in 5.7, respectively, we have the following inequalities:
G < A1/3
G < A2/3 A −1/3
1/2 G2/3 < A < A2A−1
1/2 < Z < A2A−1, 5.10
G < Z1/3
1/2 G2/3 < I < Z1/2 < Y < Z, 5.11 which contain5.3 and 5.4 Here we have used the formula Ia2, b2/Ia, b Za, b see
9, Remark 3
Example 5.2 ByCorollary 2.4, we can get another more refined inequalities Forr s > 0, we
have
2,1
2;r, s; a, b > F 2
3,1
3;r, s; a, b > F 3
4,1
4;r, s; a, b > F1, 0; r, s; a, b
> F 4
3, −1
3;r, s; a, b
> F 3
2, −1
2;r, s; a, b
> F2, −1; r, s; a, b,
5.12
that is,
Ia r/2 , b r/2
Ia s/2 , b s/2
2/r−s
> a r/3 b r/3
a s/3 b s/3
3/r−s
> a r/2
√
a r/2 b r/2 b r/2
a s/2√a s/2 b s/2 b s/2
2/r−s
> s r
a r − b r
a s − b s
1/r−s
> a r/3 b r/3
a s/3 b s/3
a2r/3 b2r/3
a2s/3 b2s/3
3/5r−s
G2/5
> a r
√
a r b r b r
a s√a s b s b s
1/2r−s√
G > a a r s b b r s
1/3r−s
G2/3 ,
5.13 which can be concisely denoted by
Ia r/2 , b r/2
Ia s/2 , b s/2
2/r−s
>
Aa r/3 , b r/3
Aa s/3 , b s/3
3/r−s
>
ha r/2 , b r/2
ha s/2 , b s/2
2/r−s
>
La r , b r
La s , b s
1/r−s
>
Aa r/3 , b r/3
Aa s/3 , b s/3A
a2r/3 , b2r/3
Aa2s/3 , b2s/3
3/5r−s
G2/5
>
ha r , b r
ha s , b s
1/2r−s
√
G >
Aa r , b r
Aa s , b s
1/3r−s
G2/3 ,
5.14 whereLx, y, Ix, y, Ax, y, and hx, y are defined by 1.2–1.5, respectively
... 95 Chains of inequalities for two-parameter means
Leta and b be positive numbers The p-order power mean, Heron mean, logarithmic... s; a, b are strictly increasing (decreasing) in either p or
q on −∞, ∞.
Trang 5Table... L r, s; x, y Namely, the monotonicity and log-convexity< /i>
of the functionHHL also depend on the signs of< i>I ln f xy < and