Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 763252, 7 pages
doi:10.1155/2009/763252
Research Article
Inequalities for Generalized Logarithmic Means
Yu-Ming Chu1 and Wei-Feng Xia2
1
2
Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
School of Teacher Education, Huzhou Teachers College, Huzhou 313000, China
Correspondence should be addressed to Yu-Ming Chu, chuyuming2005@yahoo.com.cn
Received 2 June 2009; Accepted 10 December 2009
Recommended by Wing-Sum Cheung
For p ∈ R, the generalized logarithmic mean Lp of two positive numbers a and b is defined
as Lp a, b a, for a b, LP a, b bp1 − ap1 /p 1b − a
1/p
, for a /
b, p /
− 1, p /
0,
1/b−a
b, p −1, and LP a, b 1/ebb /aa , for a /
b,
LP a, b b − a/log b − log a, for a /
p 0. In this paper, we prove that Ga, b Ha, b 2L−7/2 a, b, Aa, b Ha, b 2L−2 a, b,
and L−5 a, b Ha, b for all a, b > 0, and the constants −7/2, −2, and −5 cannot
√ be improved
for the corresponding inequalities. Here Aa, b a b/2 L1 a, b, Ga, b ab L−2 a, b,
and Ha, b 2ab/a b denote the arithmetic, geometric, and harmonic means of a and b,
respectively.
Copyright q 2009 Y.-M. Chu and W.-F. Xia. 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
For p ∈ R, the generalized logarithmic mean Lp a, b and power mean Mp a, b of two
positive numbers a and b are defined as
⎧
a,
⎪
⎪
⎪
⎪
⎪
1/p
⎪
⎪
⎪
⎪
bp1 − ap1
⎪
⎪
,
⎪
⎪
⎪
⎨ p 1b − a
Lp a, b b−a
⎪
⎪
,
⎪
⎪
log b − log a
⎪
⎪
⎪
⎪
⎪
1/b−a
⎪
⎪
⎪
1 bb
⎪
⎪
,
⎩
e aa
a b,
a/
b, p /
− 1, p /
0,
a/
b, p −1,
a/
b, p 0,
1.1
2
Journal of Inequalities and Applications
⎧
⎪
ap bp 1/p
⎪
⎨
,
2
Mp a, b ⎪
⎪
⎩√
ab,
p/
0,
1.2
p 0.
It is well known that Lp a, b and Mp a, b are continuous and increasing with respect
1/b−a
b
a
to p ∈ R for fixed a and b. Let Aa,
, La, b √ b a b/2, Ia, b 1/eb /a b − a/log b − log a, Ga, b ab, and Ha, b 2ab/a b be the arithmetic, identric,
logarithmic, geometric, and harmonic means of a and b, respectively. Then
min{a, b} Ha, b M−1 a, b Ga, b M0 a, b L−2 a, b La, b
L−1 a, b Ia, b L0 a, b Aa, b L1 a, b
1.3
M1 a, b max{a, b}.
In 1, the following results are established: 1 p 1/3 implies that La, b Mp a, b; 2 p 0 implies that La, b Mp a, b; 3 p < 1/3 implies that there exist
a, b > 0 such that La, b > Mp a, b; 4 p > 0 implies that there exist a, b > 0 such that
La, b < Mp a, b. Hence the question was answered: what are the least value q and the
greatest value p such that the inequality Mp a, b La, b Mq a, b holds for all a, b > 0?
Stolarsky 2 proved that Ia, b L0 a, b M2/3 a, b, with equality if and only if
a b.
In 3, Pittenger proved that
Mp1 a, b Lp a, b Mp2 a, b
1.4
for all a, b > 0, where
⎧
⎪
p log 2
p2
⎪
⎪
,
,
⎪min
⎪
⎪
3 log p 1
⎪
⎪
⎨
2
p1 ,
⎪
3
⎪
⎪
⎪
⎪
⎪
p2
⎪
⎪
,0 ,
⎩min
3
⎧
p log 2
p2
⎪
⎪
⎪
,
max
,
⎪
⎪
3 log p 1
⎪
⎪
⎨
p2 log 2,
⎪
⎪
⎪
⎪
⎪
p2
⎪
⎪
⎩max
,0 ,
3
p > −1, p /
0,
p 0,
p −1,
1.5
p > −1, p /
0,
p 0,
p −1.
Journal of Inequalities and Applications
3
Here p1 , p2 are sharp and equality holds only if a b or p 1, −2 or −1/2. The case
p −1 reduces to Lin’s results 1. Other generalizations of Lin’s results were given by Imoru
4.
Qi and Guo 5 established that
br1 − ar1
bδ−a
·
b−a
b δr1 − ar1
1/r
<
Ia, b
Ia, b δ
1.6
for all b > a > 0, δ > 0 and r ∈ R. The upper bound in 1.6 is the best possible.
In 6, Chu et al. established the following result:
b − La, bΨb La, b − aΨa > b − aΨ
ab
1.7
for all b > a 2, where the Ψ function is the logarithmic derivative of the gamma function.
Recently, some monotonicity results of the ratio between generalized logarithmic
means were established in 7–9.
The purpose of this paper is to answer the following questions: what are the greatest
values p and q, and the least value r such that Ga, bHa, b 2Lp a, b, Aa, bHa, b 2Lq a, b, and Ha, b Lr a, b for all a, b > 0?
2. Main Results
Theorem 2.1. Ga, b Ha, b 2L−7/2 a, b for all a, b > 0, with inequality if and only if a b,
and the constant −7/2 cannot be improved.
Proof. If a b, then from 1.1we clearly see that Ga, b Ha, b 2L−7/2 a, b 2a. Next,
we assume that a /
b and t a/b, and then elementary computations yield
5/2t5 t 1
L−7/2 a, b b 4 3 2
t t t t1
Ga, b Ha, b b
2/7
,
tt 12
,
t2 1
Ga, b Ha, b7 − 2L−7/2 a, b7
2
7 b7 t7 t 12
12 4
3
2
3 2
2 t 1 t t t t 1 − 800t t 1
7
t2 1 t4 t3 t2 t 1
4
Journal of Inequalities and Applications
b7 t7 t 12
2
7
t2 1 t4 t3 t2 t 1
× t20 14t19 93t18 − 408t17 1186t16 −2830t15 5254t14 − 8402t13 11597t12 − 13974t11
14938t10 −13974t9 11597t8 −8402t7 5254t6 −2830t5 1186t4 −408t3 93t2 14t 1
b7 t7 t 12 t − 14
2
7
t2 1 t4 t3 t2 t 1
× t16 18t15 159t14 124t13 799t12 240t11 1757t10 258t9 2248t8
258t7 1757t6 240t5 799t4 124t3 159t2 18t 1 > 0.
2.1
To prove that −7/2 is the largest number for which the inequality holds, we take 0 <
ε < 1 and 0 < x < 1, and we see that
L−7/2ε
1/7/2−ε
5 − 2εx1 x/21 x5−2ε
,
1 x , 1 1 x5−2ε − 1
2
21 x1 x/22
,
G 1 x2 , 1 H 1 x2 , 1 1 x x2 /2
7/2−ε 7/2−ε
2L−7/2ε 1 x2 , 1
− G 1 x2 , 1 H 1 x2 , 1
27/2−ε 1 x/21 x7/2−ε
fx,
7/2−ε
1 x5−2ε − 1 1 x x2 /2
2.2
2.3
where fx 5 − 2εx1 x3/2−ε 1 x x2 /27/2−ε − 1 x/26−2ε 1 x5−2ε − 1.
Making use of the Taylor expansion, we have
3 − 2ε
3 − 2ε1 − 2ε 2 3 − 2ε1 − 2ε1 2ε 3
x
x −
x o x3
fx 5 − 2εx 1 2
8
48
7 − 2ε
7 − 2ε2 2 9 − 2ε7 − 2ε5 − 2εx3 3
3
× 1
x
x x o x
2
8
48
3 − ε5 − 2ε 2 3 − ε5 − 2ε2 − ε 3
x x o x3
− 1 3 − εx 4
12
2 − ε3 − 2ε 2 2 − ε3 − 2ε1 − ε 3
× 5 − 2εx 1 2 − εx x x o x3
3
6
Journal of Inequalities and Applications
47 − 38ε 8ε2 2
2
x o x
5 − 2εx 1 5 − 2εx 4
141 − 121ε 26ε2 2
2
x o x
− 5 − 2εx 1 5 − 2εx 12
5
ε7 − 2ε5 − 2ε 3
x o x3 .
12
2.4
Equations 2.3 and 2.4 imply that for any 0 < ε < 1 there exists 0 < δ δε < 1,
such that 2L−7/2ε 1 x2 , 1 > G1 x2 , 1 H1 x2 , 1 for x ∈ 0, δ.
Theorem 2.2. Aa, b Ha, b 2L−2 a, b for all a, b > 0, with equality if and only if a b, and
the constant −2 cannot be improved.
Proof. Simple computations yield
2ab
ab
Aa, b Ha, b − 2L−2 a, b − 2 ab 2
ab
√
a−
√ 4
b
2a b
0.
2.5
Next we prove that −2 is the optimal value for which the inequality holds.
For 0 < ε < 1 and 0 < t < 1, elementary computations yield
L−2ε 1 t, 1 1 − εt1 t1−ε
1/2−ε
,
1 t1−ε − 1
2 1 t t2 /8
,
A1 t, 1 H1 t, 1 1 t/2
2L−2ε 1 t, 12−ε − A1 t, 1 H1 t, 12−ε 22−ε
ft,
1 t1−ε − 1 1 t/22−ε
2.6
2.7
where ft 1 − εt1 t1−ε 1 t/22−ε − 1 t1−ε − 11 t t2 /82−ε .
Using Taylor expansion we get
2−ε
ε1 − ε 2
2 − ε1 − ε 2
2
t o t
t
t o t2
× 1
ft 1 − εt 1 1 − εt −
2
2
8
ε
ε1 ε 2
2 − ε5 − 4ε 2
2
− 1 − εt 1 − t t o t
t o t2
× 1 2 − εt 2
6
8
6
Journal of Inequalities and Applications
10 − 19ε 9ε2 2
4 − 3ε
2
t
t o t
1 − εt 1 2
8
30 − 59ε 28ε2 2
4 − 3ε
2
t
t o t
− 1 − εt 1 2
24
ε1 − ε2 − ε 3
t o t3 .
24
2.8
Equations 2.7 and 2.8 imply that for any 0 < ε < 1 there exists 0 < δ δε < 1,
such that 2L−2ε 1 t, 1 > A1 t, 1 H1 t, 1 for t ∈ 0, δ.
Theorem 2.3. Ha, b L−5 a, b for all a, b > 0, with equality if and only if a b, and the
constant −5 cannot be improved.
Proof. Form 1.1 we clearly see that L−5 a, b Ha, b a if a b. If a / b, then simple
computations yield
L−5 a, b b
4a/b4
a/b2 1a/b 1
Ha, b b
L−5 a, b5 − Ha, b5 1/5
,
2 · a/b
,
1 a/b
2.9
4
a
4b5 a/b4
− 1 > 0.
1 a/b5 1 a/b2 b
To show that −5 is the best possible constant for which the inequality holds, let 0 < ε <
1 and 0 < t < 1, and then
5ε
H1 t, 15ε − L−5ε 1 t, 1
1 t4ε
ft,
1 t/25ε 1 t4ε − 1
2.10
where ft 1 t1 t4ε − 1 − 4 εt1 t/25ε .
Using Taylor expansion we have
4 ε3 ε 2 4 ε3 ε2 ε 3
t t o t3
ft 1 t 4 εt 2
6
5ε
5 ε4 ε 2
− 4 εt 1 t
t o t2
2
8
ε4 ε5 ε 3
t o t3 .
24
2.11
Equations 2.10 and 2.11 imply that for any 0 < ε < 1 there exists 0 < δ δε < 1,
such that H1 t, 1 > L−5ε 1 t, 1 for t ∈ 0, δ.
Journal of Inequalities and Applications
7
Remark 2.4. If p 5, then
1
p
p−1
p p
p−1
a
−
2
a
−
1
,
a
p
−
1
−
1a
1
a
ap−1 − 1 1 ap
lim p − 1 a − 1ap−1 1 ap − 2p ap ap−1 − 1 ∞.
p
L−p a, 1 − Ha, 1p a → ∞
2.12
Therefore, we cannot get inequality Ha, b Lp a, b for any p ∈ R and all a, b > 0.
Remark 2.5. It is easy to verify that Aa, b Ga, b 2L−1/2 a, b for all a, b > 0.
Acknowledgments
This research is partly supported by N S Foundation of China under Grant 60850005 and
N S Foundation of Zhejiang Province under Grants D7080080 and Y7080185.
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