Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 471096, 14 pages
doi:10.1155/2012/471096
Research Article
Inequalities between Power Means and
Convex Combinations of the Harmonic and
Logarithmic Means
Wei-Mao Qian1 and Zhong-Hua Shen2
1
2
School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, China
Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China
Correspondence should be addressed to Wei-Mao Qian, qwm661977@126.com
Received 10 October 2011; Accepted 1 December 2011
Academic Editor: Md. Sazzad Chowdhury
Copyright q 2012 W.-M. Qian and Z.-H. Shen. 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 prove that αHa, b 1 − αLa, b > M1−4α/3 a, b for α ∈ 0, 1 and all a, b > 0 with
a
/ b if and
√ only if α ∈ 1/4, 1 and αHa, b 1 − αLa, b < M1−4α/3 a, b if and only if
α ∈ 0, 3 345/80 − 11/16, and the parameter 1 − 4α/3 is the best possible in either case. Here,
0 and
Ha, b 2ab/a b, La, b a − b/log a − log b, and Mp a, b ap bp /21/p p /
√
M0 a, b ab are the harmonic, logarithmic, and pth power means of a and b, respectively.
1. Introduction
The classical logarithmic mean La, b of two positive real numbers a and b with a /
b is
defined by
La, b a−b
.
log a − log b
1.1
In the recent past, the bivariate means have been the subject of intensive research. In
particular, many remarkable inequalities for La, b can be found in the literature 1–21. It
might be surprising that the logarithmic mean has applications in physics, economics, and
even in meteorology 22–24. In 22 the authors study a variant of Jensen’s functional equation involving the logarithmic mean, which appears in a heat conduction problem. A representation of La, b as an infinite product and an iterative algorithm for computing it as the
common limit of two sequences of special geometric and arithmetic means are given in 4.
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Journal of Applied Mathematics
In 25, 26 it is shown that La, b can be expressed in terms of Gauss hypergeometric
function 2 F1 . And, in 26 the authors prove that the reciprocal of the logarithmic mean
is strictly totally positive; that is, every n × n determinant with elements 1/Lai , bi , where
0 < a1 < a2 < · · · < an and 0 < b1 < b2 < · · · < bn , is positive for all n ≥ 1.
√
a
b 1/a−b
, Aa, b a Let Ga, b ab, Ha, b 2ab/a b, Ia, b
√ 1/ea /b 1/p
p
p
p /
0 and M0 a, b ab, and Lp a, b ap1 bp1 /ap b/2, Mp a, b a b /2
bp be the geometric, harmonic, identric, arithmetic, pth power, and pth Lehmer means of two
positive numbers a and b, respectively. Then it is well known that both Mp a, b and Lp a, b
are continuous and strictly increasing with respect to p ∈ R for fixed a, b > 0 with a /
b, and
the inequalities
min{a, b} < Ha, b M−1 a, b L−1 a, b < Ga, b M0 a, b L−1/2 a, b
< La, b < Ia, b < Aa, b M1 a, b L0 a, b < max{a, b}
1.2
hold for all a, b > 0 with a /
b.
In 4, Carlson proves that the double inequality
Ga, bAa, b Ga, b
1
< La, b < Aa, b Ga, b
2
2
1.3
holds for all a, b > 0 with a /
b.
In 5, Lin finds the best possible upper and lower power bounds for the logarithmic
mean as follows:
M0 a, b < La, b < M1/3 a, b
1.4
for all a, b > 0 with a /
b.
In 9, Sándor establishes that
Ga, bIa, b < La, b < Aa, b Ga, b − Ia, b
1.5
for all a, b > 0 with a /
b.
In 27, Alzer gives the optimal Lehmer mean bounds for L, LI1/2 , and L I/2 as
follows:
L−1/3 a, b < La, b < L0 a, b,
L−1/4 a, b < La, bIa, b < L0 a, b,
1
L−1/4 a, b < La, b Ia, b < L0 a, b
2
for all a, b > 0 with a /
b.
1.6
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3
The following sharp bounds for LI1/2 and L I/2 in terms of power mean are
presented in 28:
M0 a, b <
La, bIa, b < M1/2 a, b,
Mlog 2/1log 2 a, b <
1
La, b Ia, b < M1/2 a, b
2
1.7
for all a, b > 0 with a /
b.
In 29, 30, the authors obtain the sharp bounds for the products Aα a, bL1−α a, b and
Gα a, bL1−α a, b and the sum αAa, b 1 − αLa, b in terms of power mean as follows:
M0 a, b < Aα a, bL1−α a, b < M12α/3 a, b,
M0 a, b < Gα a, bL1−α a, b < M1−α/3 a, b,
1.8
Mlog 2/log 2−log α a, b < αAa, b 1 − αLa, b < M12α/3 a, b
for any α ∈ 0, 1 and all a, b > 0 with a /
b.
In 31, Zhu presents some bounds for Ia, b in terms of Aa, b and La, b and La, b
in terms of Ga, b and Ia, b.
In 32, Chu et al. prove that the double inequality αAa, b 1 − αHa, b < P a, b <
βAa, b 1 − βHa, b holds for all a, b > 0 with a /
b if and only if α ≤ 2/π and β ≥ 5/6.
It is the aim of this paper to give the optimal power mean bounds for the convex
combination of harmonic and logarithmic means. Our main result is the following theorem.
Theorem 1.1. For α ∈ 0, 1 and all a, b > 0 with a /
b, one has
1 αHa, b 1 − αLa, b > M1−4α/3 a, b if and only if α ∈ 1/4, 1;
√
2 αHa, b 1 − αLa, b < M1−4α/3 a, b if and only if α ∈ 0, 3 345/80 − 11/16.
In particular, the parameter 1 − 4α/3 is the best possible in either case.
2. Lemmas
In order to establish our main result we need to establish four lemmas, which we present in
this section.
Lemma 2.1. Let α ∈ 1/4, 1, p 1 − 4α/3 ∈ −1, 0, and gt −4αpp 12 p 2tp−1 21 −
αp2 1 − p2 tp−2 21 − αp1 − p2 2 − ptp−3 121 − α1 − p. Then gt > 0 for t ∈ 1, ∞.
Proof. Simple computations lead to
64
1 − α2 56α2 23α 11 > 0,
81
lim gt 121 − α 1 − p 81 − α1 2α > 0,
g1 t → ∞
g t −2p 1 − p tp−4 g1 t,
2.1
2.2
2.3
4
Journal of Applied Mathematics
where
2 g1 t −2α p 1 p 2 t2 1 − αp p 1 2 − p t 1 − α 1 − p 2 − p 3 − p ,
g1 1 4
1 − α 148α2 − 11α 25 > 0,
27
lim g1 t −∞,
t → ∞
2 g 1 t −4α p 1 p 2 t 1 − αp p 1 2 − p
−
4
1 − α2 16α7 − 4αt 4α − 14α 5 < 0
27
2.4
2.5
2.6
for t ∈ 1, ∞.
Inequality 2.6 implies that g1 t is strictly decreasing in 1, ∞. Then 2.4 and 2.5
lead to the conclusion that there exists λ1 > 1 such that g1 t > 0 for t ∈ 1, λ1 and g1 t < 0
for t ∈ λ1 , ∞. It follows from 2.3 that gt is strictly increasing in 1, λ1 and strictly
decreasing in λ1 , ∞.
Therefore, Lemma 2.1 follows from 2.1 and 2.2 together with the piecewise
monotonicity of gt.
Lemma 2.2. Let α ∈ 1/4, 1, p 1 − 4α/3 ∈ −1, 0, and ht −1 − αp 1p 22 p 3tp p 1p3 − αp3 − 19αp2 3p2 − 34αp 2p − 8αtp−1 1 − αpp3 − 8p2 − p 4tp−2 1 −
α1 − pp3 5p2 − 14p 4tp−3 41 − α7 − 4p − 4p1 − αt−1 4α1 pt−2 . Then ht > 0
for t ∈ 1, ∞.
Proof. Let
h1 t t3−p ht.
2.7
Then simple computations lead to
16
1 − α 80α2 110α − 1 > 0,
27
2 h1 t −31 − α p 1 p 2 p 3 t2 2 p 1
× p3 − αp3 − 19αp2 3p2 − 34αp 2p − 8α t 1 − αp p3 − 8p2 − p 4
h1 1 2.8
41 − α 7 − 4p 3 − p t2−p − 4p1 − α 2 − p t1−p 4α 1 − p2 t−p ,
h1 1 32
1 − α −16α3 38α2 176α − 9 > 0,
27
2.9
Journal of Applied Mathematics
5
2 h1 t −61 − α p 1 p 2 p 3 t 2 p 1
× p3 − αp3 − 19αp2 3p2 − 34αp 2p − 8α
41 − α 7 − 4p 3 − p 2 − p t1−p
− 4p1 − α 2 − p 1 − p t−p − 4αp 1 − p2 t−p−1 ,
2.10
8
1 − α −128α4 896α3 288α2 5294α − 437 > 0,
81
2 p 3 41 − α 7 − 4p 3 − p 2 − p 1 − p t−p
h
1 t −61 − α p 1 p 2
h1 1 2 4p2 1 − α 2 − p 1 − p t−p−1 4αp 1 p 1 − p t−p−2 ,
h
1 1 8
1 − α 576α4 3872α3 660α2 6612α − 785 > 0,
81
4
h1 t −4p 1 − p t−p−3 h2 t,
2.11
2.12
where
2 h2 t 1 − α 7 − 4p 3 − p 2 − p t2 1 − αp 2 − p p 1 t α p 1 p 2 ,
4
1 − α 96α3 232α2 388α 175 > 0,
27
h2 t 21 − α 7 − 4p 3 − p 2 − p t 1 − αp 2 − p p 1
h2 1 ≥ h2 1 4
1 − α5 4α 12α2 31α 23 > 0
9
2.13
for t ∈ 1, ∞.
From inequalities 2.13 we clearly see that h2 t > 0 for t ∈ 1, ∞. Then 2.12 leads
to the conclusion that h
1 t is strictly increasing in 1, ∞.
Therefore, Lemma 2.2 follows from 2.7–2.11 and the monotonicity of h
1 t.
√
Lemma 2.3. Let α ∈ 0, 1, p 1 − 4α/3, λ0 3 345/80 − 11/16 0.00903 . . ., and ft 2α1 − tp1 t log2 t 1 − α1 tp−1 1 t2 t log t 1 − α1 t2 1 − ttp 1. Then the following
two statements are true:
1 if α ∈ 1/4, 1, then ft > 0 for t ∈ 1, ∞;
2 if α ∈ 0, λ0 , then ft < 0 for t ∈ 1, ∞.
6
Journal of Applied Mathematics
Proof. Let f1 t t−p f t, f2 t tp2 f1 t, f3 t t4−p f2 t, f4 t tp2 f3 t and f5 t t4−p f4 t. Then simple computations lead to
f1 0,
2.14
f t 2α 1 − p 2 tp1 log2 t
p 2 − αp − 6α tp1 21 − α p 1 tp
1 − αptp−1 31 − αt2 41 − αt 3α 1 log t
− 1 − α p 3 tp2 p 1 tp1 − p 3 tp − p 1 tp−1 2t2 − 2 ,
f 1 0,
2.15
f1 t −2α p 1 p 2 log2 t
p2 − αp2 3p − 11αp − 14α 2 21 − αp p 1 t−1
−1 − αp 1 − p t−2 61 − αt1−p 41 − αt−p 4αpt−1−p log t
− 1 − α p 2 p 3 t 1 − α p2 5p 2 t−1
1 − α p2 p − 1 t−2 − 1 − αt1−p 41 − αt−p 1 3αt−1−p
− 1 − αp2 − 1 − αp − 5α 1,
f1 1 0,
2.16
f2 t − 4α p 1 p 2 tp1 21 − αp p 1 tp − 21 − αp 1 − p tp−1
−61 − α 1 − p t2 41 − αpt 4α 1 p log t
− 1 − α p 2 p 3 tp2 p2 − αp2 3p − 11αp − 14α 2 tp1
1 − α p2 − 3p − 2 tp − 1 − α p2 3p − 2 tp−1 1 − α p 5 t2
41 − α 1 − p t α − 3αp − p − 1,
f2 1 0,
2.17
2
2 f2 t − 4α p 1 p 2 tp 21 − αp2 p 1 tp−1 21 − αp 1 − p tp−2
2 −121 − α 1 − p t 41 − αp log t − 1 − α p 2 p 3 tp1
p 1 p2 − αp2 − 15αp 3p − 22α 2 tp 1 − αp p2 − 5p − 4 tp−1
1 − α 1 − p p2 5p − 2 tp−2 41 − α 4 − p t − 4α 1 p t−1
41 − α 1 − 2p ,
f2 1 0,
2.18
Journal of Applied Mathematics
2
2 f2 t − 4αp p 1 p 2 tp−1 − 21 − αp2 1 − p2 tp−2 − 21 − αp 1 − p
7
2
× 2 − p tp−3 − 121 − α 1 − p log t − 1 − α p 1 p 2
× p 3 tp p 1 p3 − αp3 − 19αp2 3p2 − 34αp 2p − 8α tp−1
1 − αp p3 − 8p2 − p 4 tp−2 1 − α 1 − p p3 5p2 − 14p 4 tp−3
− 41 − αpt−1 4α 1 p t−2 41 − α 7 − 4p .
2.19
1 If α ∈ 1/4, 1, then from 2.19 we note that
f2 t gt log t ht,
2.20
where gt and ht are defined as in Lemmas 2.1 and 2.2, respectively.
Lemmas 2.1 and 2.2 together with 2.20 imply that f2 t is strictly increasing in
1, ∞. Therefore, ft > 0 for t ∈ 1, ∞ follows from 2.14–2.18 and the monotonicity
of f2 t.
2 If α ∈ 0, λ0 , then from 2.19 we have
16
1 − α 80α2 110α − 1
27
√
3 345 11
1280
≤ 0,
1 − αα − λ0 α 27
80
16
f2 1 2.21
2 f3 t 4αp 1 − p p 1 p 2 t2 − 21 − αp2 1 − p2 2 − p t
2 2 −21 − αp 1 − p 2 − p 3 − p log t − 1 − αp p 1 p 2 p 3 t3
p 1 p4 − αp4 − 22αp3 2p3 − 27αp2 − p2 − 2p 18αp 8α t2
1 − αp p4 − 12p3 15p2 8p − 8 t 1 − α 1 − p p4 4p3 − 35p2 50p − 12
121 − α 1 − p t3−p 4p1 − αt2−p − 8α 1 p t1−p ,
32
1 − α1 − 4α 80α2 110α − 1
81
√
3 345 11
10240
1
− α α − λ0 α ≤ 0,
1 − α
81
4
80
16
f3 1 2.22
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Journal of Applied Mathematics
2 f3 t 8αp 1 − p p 1 p 2 t − 21 − αp2 1 − p2 2 − p log t
2 − 31 − αp p 1 p 2 p 3 t − 2 p 1
× 3αp4 − p4 26αp3 − 2p3 25αp2 p2 − 22αp 2p − 8α t
2 − 1 − αp p4 8p3 − 17p2 − 4p 8 − 21 − αp 1 − p 2 − p 3 − p t−1
121 − α 1 − p 3 − p t2−p 41 − αp 2 − p t1−p − 8α 1 − p2 t−p ,
8
f3 1 1 − α 3328α4 128α3 − 7248α2 7118α − 167
243
8
<
1 − α 3328λ40 128λ30 7118λ0 − 167
243
8
<
1 − α 3328 × 0.014 128 × 0.013 7118 × 0.01 − 167 < 0,
243
2.23
2 2 f3 t 8αp 1 − p p 1 p 2 log t − 61 − αp p 1 p 2 p 3 t
2 − 21 − αp2 1 − p2 2 − p t−1 21 − αp 1 − p 2 − p 3 − p t−2
121 − α 1 − p 2 − p 3 − p t1−p 41 − αp 1 − p 2 − p t−p
8αp 1 − p2 t−1−p − 2 p 1
× 7αp4 − p4 34αp3 − 2p3 21αp2 p2 − 30αp 2p − 8α ,
8
f3 1 1 − α7 − 4α 256α4 − 64α3 − 1152α2 2066α − 53
243
8
<
1 − α7 − 4α 256λ40 2066λ0 − 53
243
8
<
1 − α7 − 4α 256 × 0.014 2066 × 0.01 − 53 < 0,
243
2 2 f4 t −61 − αp p 1 p 2 p 3 tp2 8αp 1 − p p 1 p 2 tp1
2 21 − αp2 1 − p2 2 − p tp − 41 − αp 1 − p 2 − p 3 − p tp−1
2 121 − α 1 − p 2 − p 3 − p t2 − 41 − αp2 1 − p 2 − p t
2
− 8αp 1 − p 1 p ,
8
f4 1 1 − α −1024α4 21952α3 − 10968α2 13474α − 835
243
8
<
1 − α 21952λ30 13474λ0 − 835
243
8
<
1 − α 21952 × 0.013 13474 × 0.01 − 835 < 0,
243
3 3 f4 t −61 − αp p 1 p 2 p 3 tp1 8αp 1 − p p 1 p 2 tp
3 21 − αp3 1 − p2 2 − p tp−1 41 − αp 1 − p 2 − p 3 − p tp−2
2 241 − α 1 − p 2 − p 3 − p t − 41 − αp2 1 − p 2 − p ,
2.24
2.25
Journal of Applied Mathematics
8
1 − α7 − 4α −1024α4 21952α3 − 10968α2 13474α − 835
729
8
<
1 − α7 − 4α 21952λ30 13474λ0 − 835
729
8
<
1 − α7 − 4α 21952 × 0.013 13474 × 0.01 − 835 < 0,
729
9
f4 1 2.26
2 3 3 f4 t −61 − αp p 1 p 2 p 3 tp 8αp2 1 − p p 1 p 2 tp−1
3 2 2 − 21 − αp3 1 p 1 − p 2 − p tp−2 − 41 − αp 1 − p 2 − p 3 − p tp−3
2 241 − α 1 − p 2 − p 3 − p ,
f4 1 32
1 − α −4096α6 136320α5 − 241440α4 383672α3
2187
−209850α2 100113α − 7255
32
1 − α 136320λ50 383672λ30 100113λ0 − 7255
2187
32
<
1 − α 136320 × 0.015 383672 × 0.013 100113 × 0.01 − 7255
2187
< 0,
<
2.27
2 3 2 3 f5 t −61 − αp2 p 1 p 2 p 3 t3 − 8αp2 1 − p p 1 p 2 t2
3 2 2
2 2
21 − αp3 1 p 1 − p 2 − p t 41 − αp 1 − p 2 − p 3 − p ,
32
f5 1 1 − α1 − 4α −4096α6 173568α5 − 190368α4 439136α3
6561
−191370α2 96723α − 8665
32
1 − α1 − 4α 173568λ50 439136λ30 96723λ0 − 8665
6561
32
<
1 − α1 − 4α 173568 × 0.015 439136 × 0.013 96723 × 0.01 − 8665
6561
< 0,
2.28
<
2 3 2 3 f5 t −181 − αp2 p 1 p 2 p 3 t2 − 16αp2 1 − p p 1 p 2 t
2 2
21 − αp3 1 p 1 − p 2 − p ,
32
f5 1 1 − α2 1 − 4α2 −17408α4 69920α3 − 119136α2 95282α − 30845
6561
32
<
1 − α1 − 4α2 69920λ30 95282λ0 − 30845
6561
32
<
1 − α1 − 4α2 69920 × 0.013 95282 × 0.01 − 30845
6561
< 0,
2.29
10
Journal of Applied Mathematics
2 3 2 3 f5 t −361 − αp2 p 1 p 2 p 3 t − 16αp2 1 − p p 1 p 2 ,
128
f5 1 1 − α3 1 − 4α2 7 − 4α 160α3 − 1856α2 3370α − 2205
6561
128
<
1 − α3 1 − 4α2 7 − 4α 160λ30 3370λ0 − 2205
6561
128
<
1 − α3 1 − 4α2 7 − 4α 160 × 0.013 3370 × 0.01 − 2205
6561
< 0,
2 3 f5 t −361 − αp2 p 1 p 2 p 3
128
−
5 − 2α1 − 4α2 1 − α3 7 − 4α3 < 0.
729
2.30
Inequalities 2.30 imply that f5 t is strictly decreasing in 1, ∞. Then 2.29 leads
to the conclusion that f5 t is strictly decreasing in 1, ∞.
It follows from 2.28 and the monotonicity of f5 t that f4 t is strictly decreasing in
1, ∞. Then inequalities 2.25–2.27 lead to the conclusion that f4 t < 0 for t ∈ 1, ∞.
Thus, f3 t is strictly decreasing in 1, ∞.
From inequalities 2.22–2.24 and the monotonicity of f3 t we clearly see that f3 t <
0 for t ∈ 1, ∞. Thus, f2 t is strictly decreasing in 1, ∞.
It follows from 2.17 and 2.18 and inequality 2.21 together with the monotonicity
of f2 t that f2 t < 0 for t ∈ 1, ∞, which implies that f1 t is strictly decreasing in 1, ∞.
Therefore, ft < 0 for t ∈ 1, ∞ follows from 2.14–2.16 and the monotonicity of
f1 t.
Lemma 2.4. 3t4 − 4t2t2 − t 2 log t − 3 > 0 for t > 1.
Proof. Let
Jt 3t4 − 4t 2t2 − t 2 log t − 3.
2.31
Then simple computations lead to
J1 0,
J t 4 3t3 − 2t2 t − 2 − 8 3t2 − t 1 log t,
J 1 0,
4
J t J1 t,
t
2.32
where J1 t 9t3 − 10t2 3t − 2 − 26t − 1t log t,
J1 1 0,
2.33
J1 t
27t − 32t 5 − 212t − 1 log t,
J1 1
0,
2
J1 t 2
J2 t,
t
2.34
2.35
Journal of Applied Mathematics
11
where J2 t 27t2 − 12t log t − 28t 1,
J2 1 0,
J2 t 54t − 12 log t − 40 > 0
2.36
for t > 1.
Therefore, Lemma 2.4 follows from 2.31–2.36.
3. Proof of Theorem 1.1
Proof of Theorem 1.1. For all a, b > 0 with a /
b, we first prove that
αHa, b 1 − αLa, b > M1−4α/3 a, b
3.1
αHa, b 1 − αLa, b < M1−4α/3 a, b
3.2
for α ∈ 1/4, 1,
√
for α ∈ 0, 3 345/80 − 11/16.
Without loss of generality, we assume that a > b, t a/b > 1 and p 1 − 4α/3. We
divide the proof into two cases.
√ Case 1 α 1/4. Let x t a/b > 1. Then we clearly see that
1
Ha, b 3La, b − ab
4
b 3x4 − 4x 2x2 − x 2 log x − 3
.
8x2 1 log x
αHa, b 1 − αLa, b − M1−4α/3 a, b 3.3
Therefore, inequality 3.1 follows from 3.3 and Lemma 2.4.
√
Case 2 α ∈ 0, 3 345/80 − 11/16 ∪ 1/4, 1. Then we have
αHa, b 1 − αLa, b − M1−4α/3 a, b αHa, b 1 − αLa, b − Mp a, b
p
t 1 1/p
2αt
1 − αt − 1
.
−
b
t1
log t
2
3.4
Let
p
2αt
t 1
1
1 − αt − 1
Ft log
− log
.
t1
log t
p
2
3.5
12
Journal of Applied Mathematics
Then simple computations lead to
lim Ft 0,
t→1
F t tt 1tp
ft
,
1 2αt log t 1 − αt2 − 1 log t
3.6
where ft is defined as in Lemma 2.3.
If α ∈ 1/4, 1, then inequality 3.1 follows from 3.4–3.6 and Lemma 2.31. If α ∈
√
0, 3 345/80 − 11/16, then inequality 3.2 follows from 3.4–3.6 and Lemma 2.32.
Next, we prove that the parameter 1 − 4α/3 in inequalities 3.1 and 3.2 is the best
possible.
√
0, and x > 0, one has
For any α ∈ 0, 3 345/80 − 11/16 ∪ 1/4, 1, p /
αH1, 1 x 1 − αL1, 1 x − Mp 1, 1 x 21/p 1
Qx
,
x/2 log1 x
3.7
where Qx 21/p α1x log1x21/p 1−αx1x/2−1x/2 log1x11xp 1/p .
Letting x → 0 and making use of Taylor expansion, we get
Qx 21/p
8
1 − 4α
− p x3 o x3 .
3
3.8
If α ∈ 1/4, 1 and p > 1 − 4α/3, then 3.7 and 3.8 imply that there exists δ1 δ1 α, p > 0 such that αH1, 1 x 1 − αL1, 1 x < Mp 1, 1 x for x ∈ 0, δ1 . If
√
α ∈ 0, 3 345/80 − 11/16 and p < 1 − 4α/3, then 3.7 and 3.8 imply that there exists
δ2 δ2 α, p > 0 such that αH1, 1 x 1 − αL1, 1 x > Mp 1, 1 x for x ∈ 0, δ2 .
Finally, we prove that there exist a1 , b1 , a2 , b2 > 0 with a1 / b1 and a2 / b2 such
that αHa1 , b1 1 − αLa1 , b1 < M1−4α/3 a1 , b1 and αHa2 , b2 1 − αLa2 , b2 >
√
M1−4α/3 a2 , b2 for any 3 345/80 − 11/16 < α < 1/4.
√
If 3 345/80 − 11/16 < α < 1/4, then from the expression of f2 1 in 2.21 we clearly
see that f2 1 > 0, which leads to the conclusion that there exists λ > 1 such that
f2 t > 0
3.9
for t ∈ 1, λ.
From 2.14–2.18 and inequality 3.9 we know that
ft > 0
3.10
for t ∈ 1, λ. Equations 3.4–3.6 and inequality 3.10 lead to the conclusion that αHa, b
1 − αLa, b > M1−4α/3 a, b for all a/b ∈ 1, λ.
Journal of Applied Mathematics
13
On the other hand, simple computations lead to
lim
M1−4α/3 1, x
1 − αL1, x
3/1−4α
1 x4α−1/3
3/4α−1
2
∞.
lim
x → ∞ 2α/x 1 1 − α1 − 1/x/ log x
x → ∞ αH1, x
3.11
Equation 3.11 implies that there exists X Xα > 1 such that αHa, b 1 −
αLa, b < M1−4α/3 a, b for all a/b ∈ X, ∞.
Acknowledgment
This work was supported by the Natural Science Foundation of Zhejiang Broadcast and TV
University Grant no. XKT-09G21.
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