International Journal of Analysis and Applications
ISSN 2291-8639
Volume 13, Number 2 (2017), 132-135
http://www.etamaths.com
BOUNDING THE DIFFERENCE AND RATIO BETWEEN THE WEIGHTED
ARITHMETIC AND GEOMETRIC MEANS
FENG QI1,2,3,∗
Abstract. In the paper, making use of two integral representations for the difference and ratio of the
weighted arithmetic and geometric means and employing the weighted arithmetic-geometric-harmonic
mean inequality, the author bounds the difference and ratio between the weighted arithmetic and
geometric means in the form of double inequalities.
1. Main results
In [3, Theorem 2.3, Eq. (2.16)], the difference A(a, b; λ) − G(a, b; λ) between the weighted arithmetic
mean A(a, b; λ) = λa + (1 − λ)b and the weighted geometric mean G(a, b; λ) = aλ b1−λ was expressed
as an integral representation
Z
sin(λπ) b G(t − a, b − t; λ)
dt
(1.1)
A(a, b; λ) − G(a, b; λ) =
π
t
a
A(a,b;λ)
for b > a > 0 and λ ∈ (0, 1). In [4, Remark 4.1], the ratio G(a,b;λ)
between the weighted arithmetic
mean A(a, b; λ) and the weighted geometric mean G(a, b; λ) was expressed as an integral representation
Z
sin(λπ) b G(t − a, b − t; 1 − λ)
A(a, b; λ)
=1+
dt
(1.2)
G(a, b; λ)
π
t2
a
for b > a > 0 and λ ∈ (0, 1).
In this paper, making use of the integral representations (1.1) and (1.2) and employing the weighted
arithmetic-geometric-harmonic mean inequality
A(a, b; λ) > G(a, b; λ) > H(a, b; λ)
for b > a > 0 and λ ∈ (0, 1), where H(a, b; λ) =
1
1−λ
λ
a+ b
(1.3)
, for b > a > 0 and λ ∈ (0, 1) is called the
A(a,b;λ)
weighted harmonic mean, we will bound the difference A(a, b; λ) − G(a, b; λ) and the ratio G(a,b;λ)
of the weighted arithmetic mean A(a, b; λ) and the geometric mean G(a, b; λ) in the form of double
inequalities.
Our main results can be stated as the following theorems.
Theorem 1.1. For b > a > 0 and λ ∈ (0, 1), the difference between the weighted arithmetic and
geometric means can be bounded by
sin(λπ)
b
(2λ − 1)(b − a) + [(1 − λ)b − λa] ln
> [λa + (1 − λ)b] − aλ b1−λ
π
a

λ(1 − λ)(b − a)2
1
ab(ln b − ln a)
b−a
1
sin(λπ)



ln
−
1
−
+
, λ 6= ;
2 [λb − (1 − λ)a]
π
(2λ
−
1)
λ
λb
−
(1
−
λ)a
2λ
−
1
2
>
(1.4)

1 b2 − 2ab(ln b − ln a) − a2
1


,
λ= .
π
b−a
2
2010 Mathematics Subject Classification. Primary 26E60; Secondary 26D07, 47A64.
Key words and phrases. difference; ratio; weighted arithmetic mean; weighted geometric mean; weighted harmonic
mean; integral representation; double inequality; weighted arithmetic-geometric-harmonic mean inequality.
c
2017
Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.
132
BOUNDING DIFFERENCE AND RATIO OF ARITHMETIC AND GEOMETRIC MEANS
133
Theorem 1.2. For b > a > 0 and λ ∈ (0, 1), the ratio between the weighted arithmetic and geometric
means can be bounded by
1+
sin(λπ) b2 − a2
b
a
λa + (1 − λ)b
λ
+ (1 − 2λ) ln + − 1 >
π
ab
a
b
aλ b1−λ

b
(b − a)[λa − (1 − λ)b]
sin(λπ) (1 − λ)b2 − λa2


ln +
1+


2
2

π
[λa
−
(1
−
λ)b]
a
[λa

− (1 − λ)b]

2
(1 − λ)λ(b − a)
1
1
>
+
ln
− 1 , λ 6= ;
2

(2λ
−
1)[λa
−
(1
−
λ)b]
λ
2



2 (a + b) b
1


ln − 2 , λ = .
1 +
π b−a
a
2
(1.5)
2. Proofs of Theorems 1.1 and 1.2
Now we start out to prove our main results.
Proof of Theorem 1.1. By virtue of the inequality in the left-hand side of (1.3), we obtain
Z
b
a
Z b
(t − a)λ (b − t)1−λ
λ(t − a) + (1 − λ)(b − t)
dt <
dt
t
t
a
= [(1 − λ)b − λa](ln b − ln a) + (2λ − 1)(b − a).
Substituting this into (1.1) yields
[λa + (1 − λ)b] − aλ b1−λ <
b
sin(λπ)
[(1 − λ)b − λa] ln + (2λ − 1)(b − a) .
π
a
By virtue of the inequality in the right-hand side of (1.3), we obtain
Z
a
b
(t − a)λ (b − t)1−λ
dt >
t
Z
b
a
1
t
λ
t−a
1
+
1−λ
b−t
dt
λ(1 − λ)(b − a)2
1
ab(ln b − ln a)
b−a
=
ln
−1 −
+
.
(2λ − 1)2 (λb − (1 − λ)a)
λ
λb − (1 − λ)a 2λ − 1
Substituting this into (1.1) yields
[λa + (1 − λ)b] − aλ b1−λ >
sin(λπ)
1
λ(1 − λ)(b − a)2
ln
−
1
π
(2λ − 1)2 [λb − (1 − λ)a]
λ
ab(ln b − ln a)
b−a
1 b2 − 2ab(ln b − ln a) − a2
−
+
→
λb − (1 − λ)a 2λ − 1
π
b−a
as λ → 21 . The double inequality (1.4) is thus proved. The proof of Theorem 1.1 is complete.
Proof of Theorem 1.2. By virtue of the inequality in the left-hand side of (1.3), we obtain
Z
a
b
Z b
(t − a)1−λ (b − t)λ
(1 − λ)(t − a) + λ(b − t)
d
t
<
dt
t2
t2
a
b2 − a 2
b
a
=λ
+ (1 − 2λ) ln + − 1.
ab
a
b
Substituting this into (1.2) yields
λa + (1 − λ)b
sin(λπ) b2 − a2
b
a
−
1
<
λ
+
(1
−
2λ)
ln
+
−
1
.
aλ b1−λ
π
ab
a
b
134
F. QI
By virtue of the inequality in the right-hand side of (1.3), we obtain
Z b
Z b
(t − a)1−λ (b − t)λ
1
1
d
t
>
dt
λ
2
2 1−λ
t
t
+
a
a
t−a
b−t
(1 − 2λ) λa2 − (1 − λ)b2 ln ab + (b − a) (2λ − 1)[λa − (1 − λ)b] + (1 − λ)λ(b − a) ln
=
(2λ − 1)[λa − (1 − λ)b]2
2(a + b) b
ln − 4
→
b−a
a
1
λ
−1
as λ → 21 . Substituting this into (1.2) yields
λa + (1 − λ)b
sin(λπ) (1 − λ)b2 − λa2
b
(b − a)[λa − (1 − λ)b]
−1>
ln +
λ
1−λ
2
a b
π
[λa − (1 − λ)b]
a
[λa − (1 − λ)b]2
2
1
(1 − λ)λ(b − a)
2 (a + b) b
ln
+
−
1
→
ln
−
2
(2λ − 1)[λa − (1 − λ)b]2
λ
π b−a
a
as λ → 21 . The double inequality (1.5) is thus proved. The proof of Theorem 1.2 is complete.
3. Remarks
Finally we list several remarks on our main results.
Remark 3.1. When λ = 12 and b > a > 0, the double inequality (1.4) can be written as
a+b √
ln b − ln a
1 b−a b
2 a+b
ln
>
− ab >
− ab
> 0.
π
2
a
2
π
2
b−a
When λ =
1
2
and b > a > 0, the double inequality (1.5) can be written as
b−a a+b 1
2 (a + b) b
a+b
−
ln − 2 > 0.
> √ −1>
π
2ab
b
π b−a
a
2 ab
Remark 3.2. From the integral representations (1.1) and (1.2), we can easily see that all inequalities
for bounding the (weighted) geometric mean can be used to construct inequalities for bounding the
difference and ratio between the (weighted) arithmetic and geometric means.
Remark 3.3. Let 0 < ak < ak+1 for 1 ≤ k ≤ n − 1, wk > 0 for 1 ≤ k ≤ n, and z ∈QC \ [−an , −a1 ].
n
Theorem 3.1 in [7] states that the principal branch of the weighted geometric mean k=1 (z + ak )wk
has the integral representation
" `
! #Z
n
n
n−1
n
a`+1 Y
Y
X
X
1X
dt
wk
.
(3.1)
(z + ak ) =
wk ak + z −
sin
wj π
|ak − t|wk
π
t+z
a`
j=1
k=1
k=1
`=1
k=1
By the same arguments asP
in proofs of Theorems
1.1 and 1.2, we can derive from (3.1) lowerP
and upper
Qn
n
n
k
bounds for the differenceQ k=1 wk ak − k=1 aw
between
the
weighted
arithmetic
mean
k=1 wk ak
k
n
wk
and the geometric mean k=1 ak .
Remark
3.4. In [4], it was obtained that, for ak < ak+1 , z ∈ C \ [−an , −a1 ], and
Pn
Qnwk > 0 with
wk
w
=
1,
the
principal
branch
of
the
reciprocal
of
the
weight
geometric
mean
k=1 k
k=1 (z + ak )
can be represented by
!Z
n−1
`
a`+1
X
1
1
1X
1
Qn
Qn
=
sin
π
w
d t.
(3.2)
k
wk
wk t + z
π
(z
+
a
)
|t
−
a
|
k
k
a`
k=1
k=1
`=1
k=1
The integral representation (3.2) generalizes corresponding results in [2, 3] and [5, Lemma 2.4].
Remark 3.5. This paper is a companion of the articles [1–4, 6–10].
BOUNDING DIFFERENCE AND RATIO OF ARITHMETIC AND GEOMETRIC MEANS
135
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1 Institute
of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China
2 College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia
Autonomous Region, 028043, China
3 Department
of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387,
China
∗ Corresponding
author: [email protected], [email protected], [email protected]