Journal of Inequalities and Applications Volume 2009, Article ID 763252,7pages doi:10.1155/2009/763252
Research Article
Inequalities for Generalized Logarithmic Means
Yu-Ming Chu
1and Wei-Feng Xia
21Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China 2School of Teacher Education, Huzhou Teachers College, Huzhou 313000, China
Correspondence should be addressed to Yu-Ming Chu,[email protected]
Received 2 June 2009; Accepted 10 December 2009
Recommended by Wing-Sum Cheung
Forp ∈ R, the generalized logarithmic mean Lp of two positive numbers a and bis defined asLpa, b a, fora b, LPa, b bp1−ap1/p1b−a1/p, for a /b,p / −1,p /0,
LPa, b b−a/logb−loga, fora /b,p −1, andLPa, b 1/ebb/aa1/b−a, fora /b,
p0. In this paper, we prove thatGa, b Ha, b2L−7/2a, b, Aa, b Ha, b2L−2a, b,
andL−5a, b Ha, bfor alla, b >0, and the constants−7/2,−2, and−5 cannot be improved
for the corresponding inequalities. HereAa, b ab/2L1a, b, Ga, b √
abL−2a, b,
andHa, b 2ab/abdenote the arithmetic, geometric, and harmonic means of aand b, respectively.
Copyrightq2009 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 Lpa, b and power mean Mpa, b of two positive numbersaandbare defined as
Lpa, b
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
a, ab,
bp1−ap1
p1b−a
1/p
, a /b, p / −1, p /0,
b−a
logb−loga, a /b, p−1,
1 e
bb aa
1/b−a
, a /b, p0,
Mpa, b ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
apbp
2
1/p
, p /0,
√
ab, p0.
1.2
It is well known thatLpa, bandMpa, bare continuous and increasing with respect top ∈ Rfor fixedaandb. LetAa, b ab/2,Ia, b 1/ebb/aa1/b−a,La, b
b−a/logb−loga,Ga, b √ab, andHa, b 2ab/abbe the arithmetic, identric, logarithmic, geometric, and harmonic means ofaandb, respectively. Then
min{a, b}Ha, b M−1a, bGa, b M0a, b L−2a, bLa, b
L−1a, bIa, b L0a, bAa, b L1a, b
M1a, bmax{a, b}.
1.3
In 1, the following results are established: 1 p 1/3 implies that La, b Mpa, b;2p 0 implies thatLa, b Mpa, b; 3 p < 1/3 implies that there exist a, b > 0 such thatLa, b > Mpa, b;4 p > 0 implies that there exist a, b > 0 such that La, b < Mpa, b. Hence the question was answered: what are the least valueqand the greatest valuepsuch that the inequalityMpa, bLa, bMqa, bholds for alla, b >0?
Stolarsky2proved thatIa, b L0a, b M2/3a, b, with equality if and only if ab.
In3, Pittenger proved that
Mp1a, bLpa, bMp2a, b 1.4
for alla, b >0, where
p1 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ min
p2 3 ,
plog 2 logp1
, p >−1, p /0,
2
3, p0,
min
p2
3 ,0
, p−1,
p2 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ max
p2 3 ,
plog 2 logp1
, p >−1, p /0,
log 2, p0,
max
p2
3 ,0
, p−1.
Herep1,p2 are sharp and equality holds only ifa borp 1,−2 or−1/2. The case p−1 reduces to Lin’s results1. Other generalizations of Lin’s results were given by Imoru
4.
Qi and Guo5established that
bδ−a b−a ·
br1−ar1
bδr1−ar1 1/r
< IIa, ba, bδ 1.6
for allb > a >0,δ >0 andr∈R. The upper bound in1.6is the best possible. In6, Chu et al. established the following result:
b−La, bΨb La, b−aΨa>b−aΨab 1.7
for allb > a2, where theΨfunction is the logarithmic derivative of the gamma function. Recently, some monotonicity results of the ratio between generalized logarithmic means were established in7–9.
The purpose of this paper is to answer the following questions: what are the greatest valuespandq, and the least valuersuch thatGa, bHa, b2Lpa, b,Aa, bHa, b 2Lqa, b, andHa, bLra, bfor alla, b >0?
2. Main Results
Theorem 2.1. Ga, b Ha, b2L−7/2a, bfor alla, b >0, with inequality if and only ifab, and the constant−7/2cannot be improved.
Proof. Ifab, then from1.1we clearly see thatGa, b Ha, b 2L−7/2a, b 2a.Next, we assume thata /bandta/b, and then elementary computations yield
L−7/2a, b b
5/2t5t1 t4t3t2t1
2/7 ,
Ga, b Ha, b bttt212 1 ,
Ga, b Ha, b7−2L
−7/2a, b7
b7t7t12
t217t4t3t2t12
t112t4t3t2t12−800t3t217
b7t7t12
t217t4t3t2t12
×t2014t1993t18−408t171186t16−2830t155254t14−8402t1311597t12−13974t11
14938t10−13974t911597t8−8402t75254t6−2830t51186t4−408t393t214t1
b7t7t12t−14 t217t4t3t2t12
×t1618t15159t14124t13799t12240t111757t10258t92248t8
258t71757t6240t5799t4124t3159t218t1>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
1x2,1
5−2εx1x/21x5−2ε
1x5−2ε−1
1/7/2−ε ,
G1x2,1H1x2,1 21x1x/2 2
1xx2/2 ,
2.2
2L−7/2ε
1x2,17/2−ε−G1x2,1H1x2,17/2−ε
27/2−ε1x/21x7/2−ε
1x5−2ε−11xx2/27/2−εfx,
2.3
wherefx 5−2εx1x3/2−ε1xx2/27/2−ε−1x/26−2ε1x5−2ε−1. Making use of the Taylor expansion, we have
fx 5−2εx
1 3−2ε 2 x
3−2ε1−2ε 8 x
2−3−2ε1−2ε12ε 48 x
3ox3
×
17−2ε 2 x
7−2ε2 8 x
29−2ε7−2ε5−2εx3 48 x
3ox3
−
1 3−εx3−ε5−2ε 4 x
23−ε5−2ε2−ε 12 x
3ox3
×5−2εx
1 2−εx2−ε3−2ε 3 x
22−ε3−2ε1−ε 6 x
5−2εx
1 5−2εx47−38ε8ε 2
4 x
2ox2
−5−2εx
1 5−2εx141−121ε26ε 2
12 x
2ox2
ε7−2ε5−2ε
12 x
3ox3.
2.4
Equations2.3and2.4imply that for any 0< ε < 1 there exists 0 < δ δε <1, such that 2L−7/2ε1x2,1> G1x2,1 H1x2,1forx∈0, δ.
Theorem 2.2. Aa, b Ha, b2L−2a, bfor alla, b >0, with equality if and only ifab, and the constant−2cannot be improved.
Proof. Simple computations yield
Aa, b Ha, b−2L−2a, b a2b a2abb−2
ab
√
a−√b4
2ab 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−2ε1t,1
1−εt1t1−ε
1t1−ε−1
1/2−ε ,
A1t,1 H1t,1 2
1tt2/8 1t/2 ,
2.6
2L−2ε1t,12−ε−A1t,1 H1t,12−ε 2 2−ε
1t1−ε−11t/22−εft, 2.7
whereft 1−εt1t1−ε1t/22−ε−1t1−ε−11tt2/82−ε. Using Taylor expansion we get
ft 1−εt
1 1−εt−ε1−ε 2 t
2ot2×
12−ε 2 t
2−ε1−ε 8 t
2ot2
−1−εt
1−ε 2t
ε1ε 6 t
2ot2×
1 2−εt2−ε5−4ε 8 t
1−εt
14−3ε 2 t
10−19ε9ε2 8 t
2ot2
−1−εt
14−3ε 2 t
30−59ε28ε2 24 t
2ot2
ε1−ε2−ε
24 t
3ot3. 2.8
Equations2.7and2.8imply that for any 0< ε < 1 there exists 0 < δ δε <1, such that 2L−2ε1t,1> A1t,1 H1t,1fort∈0, δ.
Theorem 2.3. Ha, b L−5a, bfor all a, b > 0, with equality if and only if a b, and the constant−5cannot be improved.
Proof. Form1.1we clearly see thatL−5a, b Ha, b aifa b. Ifa /b, then simple computations yield
L−5a, b b
4a/b4
a/b21a/b1
1/5 ,
Ha, b b2·a/b 1a/b,
L−5a, b5−Ha, b5 4b 5a/b4
1a/b51 a/b2
a
b −1
4 >0.
2.9
To show that−5 is the best possible constant for which the inequality holds, let 0< ε < 1 and 0< t <1, and then
H1t,15ε−L−5ε1t,1
5ε 1t4ε
1t/25ε1t4ε−1ft, 2.10
whereft 1t1t4ε−1−4εt1t/25ε. Using Taylor expansion we have
ft 1t
4εt4ε3ε 2 t
24ε3ε2ε 6 t
3ot3
−4εt
15ε 2 t
5ε4ε 8 t
2ot2
ε4ε5ε
24 t
3ot3.
2.11
Remark 2.4. Ifp5, then
L−pa,1p−Ha,1p ap−1− 1
11ap
p−
1a−1ap−11ap−2papap−1−1,
lim a→∞
p−1a−1ap−11ap−2papap−1−1 ∞.
2.12
Therefore, we cannot get inequalityHa, bLpa, bfor anyp∈Rand alla, b >0.
Remark 2.5. It is easy to verify thatAa, b Ga, b 2L−1/2a, bfor alla, 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.
References
1 T. P. Lin, “The power mean and the logarithmic mean,”The American Mathematical Monthly, vol. 81, pp. 879–883, 1974.
2 K. B. Stolarsky, “The power and generalized logarithmic means,”The American Mathematical Monthly, vol. 87, no. 7, pp. 545–548, 1980.
3 A. O. Pittenger, “Inequalities between arithmetic and logarithmic means,” Univerzitet u Beogradu. Publikacije Elektrotehniˇckog Fakulteta. Serija Matematika i Fizika, no. 678–715, pp. 15–18, 1980.
4 C. O. Imoru, “The power mean and the logarithmic mean,”International Journal of Mathematics and Mathematical Sciences, vol. 5, no. 2, pp. 337–343, 1982.
5 F. Qi and B.-N. Guo, “An inequality between ratio of the extended logarithmic means and ratio of the exponential means,”Taiwanese Journal of Mathematics, vol. 7, no. 2, pp. 229–237, 2003.
6 Y. M. Chu, X. M. Zhang, and T. M. Tang, “An elementary inequality for psi function,”Bulletin of the Institute of Mathematics. Academia Sinica, vol. 3, no. 3, pp. 373–380, 2008.
7 X. Li, C.-P. Chen, and F. Qi, “Monotonicity result for generalized logarithmic means,”Tamkang Journal of Mathematics, vol. 38, no. 2, pp. 177–181, 2007.
8 F. Qi, S.-X. Chen, and C.-P. Chen, “Monotonicity of ratio between the generalized logarithmic means,”
Mathematical Inequalities & Applications, vol. 10, no. 3, pp. 559–564, 2007.