Best Possible Bounds for Yang Mean Using Generalized Logarithmic Mean
Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2016, Article ID 8901258, 7 pages
http://dx.doi.org/10.1155/2016/8901258
Research Article
Best Possible Bounds for Yang Mean Using Generalized
Logarithmic Mean
Wei-Mao Qian1 and Yu-Ming Chu2
1
School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, China
Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
2
Correspondence should be addressed to Yu-Ming Chu;
Received 23 January 2016; Revised 16 March 2016; Accepted 28 March 2016
Academic Editor: Kishin Sadarangani
Copyright © 2016 W.-M. Qian and Y.-M. Chu. 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 the double inequality 𝐿 𝑝 (𝑎, 𝑏) < 𝑈(𝑎, 𝑏) < 𝐿 𝑞 (𝑎, 𝑏) holds for all 𝑎, 𝑏 > 0 with 𝑎 ≠ 𝑏 if and only if 𝑝 ≤ 𝑝0 and 𝑞 ≥ 2 and
find several sharp inequalities involving the trigonometric, hyperbolic, and inverse trigonometric functions, where 𝑝0 = 0.5451 ⋅ ⋅ ⋅
is the unique solution of the equation (𝑝 + 1)1/𝑝 = √2𝜋/2 on the interval (0, ∞), 𝑈(𝑎, 𝑏) = (𝑎 − 𝑏)/[√2 arctan((𝑎 − 𝑏)/√2𝑎𝑏)], and
𝐿 𝑝 (𝑎, 𝑏) = [(𝑎𝑝+1 − 𝑏𝑝+1 )/((𝑝 + 1)(𝑎 − 𝑏))]1/𝑝 (𝑝 ≠ −1, 0), 𝐿 −1 (𝑎, 𝑏) = (𝑎 − 𝑏)/(log 𝑎 − log 𝑏) and 𝐿 0 (𝑎, 𝑏) = (𝑎𝑎 /𝑏𝑏 )1/(𝑎−𝑏) /𝑒 are the
Yang, and 𝑝th generalized logarithmic means of 𝑎 and 𝑏, respectively.
1. Introduction
Stolarsky [1] proved that the inequality
For 𝑝 ∈ R and 𝑎, 𝑏 > 0 with 𝑎 ≠ 𝑏, the 𝑝th generalized
logarithmic mean 𝐿 𝑝 (𝑎, 𝑏) is defined by
𝑎𝑝+1 − 𝑏𝑝+1
{
{
]
[
{
{
{
(𝑝 + 1) (𝑎 − 𝑏)
{
{
{
𝑎 1/(𝑎−𝑏)
𝐿 𝑝 (𝑎, 𝑏) = { 1 ( 𝑎 )
,
{
𝑏
{
𝑒 𝑏
{
{
{
𝑎−𝑏
{
{
,
{ log 𝑎 − log 𝑏
1/𝑝
, 𝑝 ≠ 0, −1,
𝑝 = 0,
(1)
𝑝 = −1.
It is well known that 𝐿 𝑝 (𝑎, 𝑏) is continuous and strictly
increasing with respect to 𝑝 ∈ R for fixed 𝑎, 𝑏 > 0 with
𝑎 ≠ 𝑏. Many classical bivariate means are the special case
of the generalized logarithmic mean. For example, 𝐺(𝑎, 𝑏) =
√𝑎𝑏 = 𝐿 −2 (𝑎, 𝑏) is the geometric mean, 𝐿(𝑎, 𝑏) = (𝑎 −
𝑏)/(log 𝑎−log 𝑏) = 𝐿 −1 (𝑎, 𝑏) is the logarithmic mean, 𝐼(𝑎, 𝑏) =
(𝑎𝑎 /𝑏𝑏 )1/(𝑎−𝑏) /𝑒 = 𝐿 0 (𝑎, 𝑏) is the identric mean, and 𝐴(𝑎, 𝑏) =
(𝑎 + 𝑏)/2 = 𝐿 1 (𝑎, 𝑏) is the arithmetic mean. Recently,
the generalized logarithmic mean has been the subject of
intensive research.
𝐿 𝑝 (𝑎, 𝑏) < 𝑀(2+𝑝)/3 (𝑎, 𝑏)
(2)
holds for all 𝑎, 𝑏 > 0 with 𝑎 ≠ 𝑏 and 𝑝 ∈ (−2, −1/2) ∪ (1, ∞),
and inequality (2) is reversed for 𝑝 ∈ (−∞, −2) ∪ (−1/2, 1),
where 𝑀𝑟 (𝑎, 𝑏) = [(𝑎𝑟 + 𝑏𝑟 )/2]1/𝑟 (𝑟 ≠ 0) and 𝑀0 (𝑎, 𝑏) = √𝑎𝑏
is the 𝑟th power mean of 𝑎 and 𝑏.
Yang [2] proved that the double inequality
𝐴 (𝑎, 𝑏) < 𝐿 𝑝 (𝑎, 𝑏) < 𝑀𝑝 (𝑎, 𝑏)
(3)
holds for all 𝑎, 𝑏 > 0 with 𝑎 ≠ 𝑏 if 𝑝 > 1, and inequality (3) is
reversed if 𝑝 < 0.
In [3], the authors proved that the inequality
𝐿 𝑝 (𝑎, 𝑏) <
𝑎+𝑏
(𝑝 + 1)
1/𝑝
(4)
holds for all 𝑎, 𝑏 > 0 with 𝑎 ≠ 𝑏 and 𝑝 > 1.
Li et al. [4] proved that the function 𝑝 → 𝐿 𝑝 (𝑎, 𝑏)/𝐿 𝑝 (1 −
𝑎, 1 − 𝑏) is strictly increasing (decreasing) on R if 0 < 𝑎 < 𝑏 ≤
1/2 (1/2 ≤ 𝑎 < 𝑏 < 1). In [5, 6], the authors proved that the
function 𝑞 → 𝐿 𝑞 (𝑎, 𝑏)/𝐿 𝑞 (𝑎, 𝑐) is strictly decreasing on R if
0 < 𝑎 < 𝑏 < 𝑐 and the function 𝑟 → 𝐿 𝑟 (𝑑, 𝑑 + 𝜀)/𝐿 𝑟 (𝑑 + 𝛿, 𝑑 +
𝜀 + 𝛿) is strictly increasing on R for all 𝑑, 𝜀, 𝛿 > 0.
2
Mathematical Problems in Engineering
Shi and Wu [7] proved that the double inequality
(𝜆𝑏 + ((1 − 𝜆) /2) (𝑎 + 𝑏))𝑝+1 − (𝜆𝑎 + ((1 − 𝜆) /2) (𝑎 + 𝑏))𝑝+1
]
[
𝜆 (𝑝 + 1) (𝑏 − 𝑎)
for all 𝑏 > 𝑎 > 𝑐 > 0 and 0 < 𝜆 < 1 if 𝑝 > 1, and inequality
(5) is reversed if 𝑝 ∈ (−1, 0) ∪ (0, 1).
Long and Chu [8] and Matejı́čka [9] presented the best
possible parameters 𝑝 = 𝑝(𝛼) and 𝑞 = 𝑞(𝛼) such that the
double inequality
𝐿 𝑝 (𝑎, 𝑏) < 𝛼𝐴 (𝑎, 𝑏) + (1 − 𝛼) 𝐺 (𝑎, 𝑏) < 𝐿 𝑞 (𝑎, 𝑏)
(6)
holds for all 𝑎, 𝑏 > 0 with 𝑎 ≠ 𝑏 and 𝛼 ∈ (0, 1/2) ∪ (1/2, 1).
In [10], Qian and Long answered the question: what are
the greatest value 𝑝 and the least value 𝑞 such that the double
inequality
𝐿 𝑝 (𝑎, 𝑏) < 𝐺𝛼 (𝑎, 𝑏) 𝐻1−𝛼 (𝑎, 𝑏) < 𝐿 𝑞 (𝑎, 𝑏)
(7)
holds for all 𝑎, 𝑏 > 0 with 𝑎 ≠ 𝑏 and 𝛼 ∈ (0, 1), where
𝐻(𝑎, 𝑏) = 2𝑎𝑏/(𝑎 + 𝑏) is the harmonic mean of 𝑎 and 𝑏.
In [11, 12], the authors proved that the double inequalities
𝐿 𝑝1 (𝑎, 𝑏) < 𝑀 (𝑎, 𝑏) < 𝐿 𝑞1 (𝑎, 𝑏) ,
𝐿 𝑝2 (𝑎, 𝑏) < 𝑇 (𝑎, 𝑏) < 𝐿 𝑞2 (𝑎, 𝑏)
(8)
hold for all 𝑎, 𝑏 > 0 with 𝑎 ≠ 𝑏 if and only if 𝑝1 ≤ 𝑝1∗ ,
𝑞1 ≥ 2, 𝑝2 ≤ 3, 𝑞2 ≥ 𝑞2∗ , where 𝑝1∗ = 1.843 ⋅ ⋅ ⋅ is the
unique solution of the equation (𝑝 + 1)1/𝑝 = 2 log(1 + √2)
on the interval (0, ∞), 𝑞2∗ = 3.152 ⋅ ⋅ ⋅ is the unique solution
of the equation (𝑞 + 1)1/𝑞 = 𝜋/2 on the interval (0, ∞),
𝑀(𝑎, 𝑏) = (𝑎 − 𝑏)/[2 sinh−1 ((𝑎 − 𝑏)/(𝑎 + 𝑏))] is the NeumanSándor mean, and 𝑇(𝑎, 𝑏) = (𝑎 − 𝑏)/[2 arctan((𝑎 − 𝑏)/(𝑎 + 𝑏))]
is the second Seiffert mean.
In [13, 14], the authors presented the best possible parameters 𝑝1 = 𝑝1 (𝑞), 𝑝2 = 𝑝2 (𝑞), 𝜆 = 𝜆(𝛼), and 𝜇 = 𝜇(𝛼) such
that the double inequalities
1/𝑞
𝐿 𝑝1 (𝑎, 𝑏) < [𝐿 (𝑎𝑞 , 𝑏𝑞 )]
< 𝐿 𝑝2 (𝑎, 𝑏) ,
𝐴 (𝑎, 𝑏) + 𝐺 (𝑎, 𝑏) 1−𝛼
𝐿 𝜆 (𝑎, 𝑏) < 𝐺𝛼 (𝑎, 𝑏) [
]
2
(9)
1/𝑝
(𝑎 + 𝑏 − 𝑐)𝑝+1 − 𝑐𝑝+1
]
< 𝐿 𝑝 (𝑎, 𝑏) < [
(𝑝 + 1) (𝑎 + 𝑏 − 2𝑐)
(5)
Very recently, Yang [16] introduced the Yang mean
𝑈 (𝑎, 𝑏) =
𝑎−𝑏
√2 arctan ((𝑎 − 𝑏) /√2𝑎𝑏)
(11)
of two distinct positive real numbers 𝑎 and 𝑏 and proved that
the inequalities
𝑃 (𝑎, 𝑏) < 𝑈 (𝑎, 𝑏) < 𝑇 (𝑎, 𝑏) ,
𝐺 (𝑎, 𝑏) 𝑇 (𝑎, 𝑏)
𝑃 (𝑎, 𝑏) 𝑄 (𝑎, 𝑏)
< 𝑈 (𝑎, 𝑏) <
,
𝐴 (𝑎, 𝑏)
𝐴 (𝑎, 𝑏)
𝑄1/2 (𝑎, 𝑏) [
2𝐺 (𝑎, 𝑏) + 𝑄 (𝑎, 𝑏) 1/2
] < 𝑈 (𝑎, 𝑏)
3
< 𝑄2/3 (𝑎, 𝑏) [
𝐺 (𝑎, 𝑏) + 𝑄 (𝑎, 𝑏) 1/3
] ,
2
(12)
𝐺 (𝑎, 𝑏) + 𝑄 (𝑎, 𝑏)
< 𝑈 (𝑎, 𝑏)
2
2 𝐺 (𝑎, 𝑏) + 𝑄 (𝑎, 𝑏) 1/2 1 1/2
) + 𝑄 (𝑎, 𝑏)]
<[ (
3
2
3
2
hold for all 𝑎, 𝑏 > 0 with 𝑎 ≠ 𝑏, where 𝑄(𝑎, 𝑏) = √(𝑎2 + 𝑏2 )/2
is the quadratic mean of 𝑎 and 𝑏.
The Yang mean 𝑈(𝑎, 𝑏) is the special case of the Seiffert
type mean 𝑇𝑀,𝑞 (𝑎, 𝑏) = (𝑎 − 𝑏)/[𝑞 arctan((𝑎 − 𝑏)/(𝑞𝑀(𝑎, 𝑏)))]
defined by Toader in [17], where 𝑀(𝑎, 𝑏) is a bivariate mean
and 𝑞 is a positive real number. Indeed, 𝑈(𝑎, 𝑏) = 𝑇𝐺,√2 (𝑎, 𝑏).
In [18, 19], the authors proved that the double inequalities
2 𝐺 (𝑎, 𝑏) + 𝑄 (𝑎, 𝑏) 𝑝 1 𝑝
[ (
) + 𝑄 (𝑎, 𝑏)]
3
2
3
1/𝑝
2 𝐺 (𝑎, 𝑏) + 𝑄 (𝑎, 𝑏) 𝑞 1 𝑞
<[ (
) + 𝑄 (𝑎, 𝑏)]
3
2
3
< 𝑈 (𝑎, 𝑏)
1/𝑞
,
21−𝜆 (𝐺 (𝑎, 𝑏) + 𝑄 (𝑎, 𝑏))𝜆 𝑄 (𝑎, 𝑏) + 𝐺 (𝑎, 𝑏) 𝑄𝜆 (𝑎, 𝑏)
< 𝐿 𝜇 (𝑎, 𝑏)
21−𝜆 (𝐺 (𝑎, 𝑏) + 𝑄 (𝑎, 𝑏))𝜆 + 𝑄𝜆 (𝑎, 𝑏)
hold for all 𝑎, 𝑏 > 0 with 𝑎 ≠ 𝑏, 𝑞 > 0 with 𝑞 ≠ 1 and 𝛼 ∈
(0, 2/3) ∪ (2/3, 1).
Gao et al. [15] provided the greatest value 𝛼 and the least
value 𝛽 such that the double inequality
𝐿 𝛼 (𝑎, 𝑏) < 𝑃 (𝑎, 𝑏) < 𝐿 𝛽 (𝑎, 𝑏)
1/𝑝
(10)
holds for all 𝑎, 𝑏 > 0 with 𝑎 ≠ 𝑏, where 𝑃(𝑎, 𝑏) = (𝑎 −
𝑏)/[2 arcsin((𝑎 − 𝑏)/(𝑎 + 𝑏))] is the first Seiffert mean of 𝑎 and
𝑏.
(13)
< 𝑈 (𝑎, 𝑏)
<
21−𝜇 (𝐺 (𝑎, 𝑏) + 𝑄 (𝑎, 𝑏))𝜇 𝑄 (𝑎, 𝑏) + 𝐺 (𝑎, 𝑏) 𝑄𝜇 (𝑎, 𝑏)
,
21−𝜇 (𝐺 (𝑎, 𝑏) + 𝑄 (𝑎, 𝑏))𝜇 + 𝑄𝜇 (𝑎, 𝑏)
𝑀𝛼 (𝑎, 𝑏) < 𝑈 (𝑎, 𝑏) < 𝑀𝛽 (𝑎, 𝑏)
hold for all 𝑎, 𝑏 > 0 with 𝑎 ≠ 𝑏 if and only if 𝑝 ≤ 𝑝0 , 𝑞 ≥ 1/5,
𝜆 ≥ 1/5, 𝜇 ≤ 𝑝1 , 𝛼 ≤ 2 log 2/(2 log 𝜋 − log 2), and 𝛽 ≥ 4/3,
where 𝑝0 = 0.1941 ⋅ ⋅ ⋅ is the unique solution of the equation
Mathematical Problems in Engineering
3
𝑝 log(2/𝜋)−log(1+21−𝑝 )+log 3 = 0 on the interval (1/10, (...truncated)
