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) = aλb1−λ was expressed
as an integral representation

A(a,b; λ) −G(a,b; λ) =
sin(λπ)

π

∫ b
a

G(t−a,b− t; λ)
t

d t (1.1)

for b > a > 0 and λ ∈ (0, 1). In [4, Remark 4.1], the ratio A(a,b;λ)
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

A(a,b; λ)

G(a,b; λ)
= 1 +

sin(λπ)

π

∫ b
a

G(t−a,b− t; 1 −λ)
t2

d t (1.2)

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) (1.3)

for b > a > 0 and λ ∈ (0, 1), where H(a,b; λ) = 1λ
a
+ 1−λ

b

, for b > a > 0 and λ ∈ (0, 1) is called the

weighted harmonic mean, we will bound the difference A(a,b; λ) − G(a,b; λ) and the ratio A(a,b;λ)
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)

π

(
(2λ− 1)(b−a) + [(1 −λ)b−λa] ln

b

a

)
> [λa + (1 −λ)b] −aλb1−λ

>




sin(λπ)

π

[
λ(1 −λ)(b−a)2

(2λ− 1)2[λb− (1 −λ)a]
ln

(
1

λ
− 1
)
−
ab(ln b− ln a)
λb− (1 −λ)a

+
b−a

2λ− 1

]
, λ 6=

1

2
;

1

π

b2 − 2ab(ln b− ln a) −a2

b−a
, λ =

1

2
.

(1.4)

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)

π

[
λ
b2 −a2

ab
+ (1 − 2λ) ln

b

a
+
a

b
− 1
]
>
λa + (1 −λ)b

aλb1−λ

>




1 +
sin(λπ)

π

{
(1 −λ)b2 −λa2

[λa− (1 −λ)b]2
ln
b

a
+

(b−a)[λa− (1 −λ)b]
[λa− (1 −λ)b]2

+
(1 −λ)λ(b−a)2

(2λ− 1)[λa− (1 −λ)b]2
ln

(
1

λ
− 1
)}

, λ 6=
1

2
;

1 +
2

π

[
(a + b)

b−a
ln
b

a
− 2
]
, λ =

1

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∫ b
a

(t−a)λ(b− t)1−λ

t
d t <

∫ b
a

λ(t−a) + (1 −λ)(b− t)
t

d t

= [(1 −λ)b−λa](ln b− ln a) + (2λ− 1)(b−a).

Substituting this into (1.1) yields

[λa + (1 −λ)b] −aλb1−λ <
sin(λπ)

π

{
[(1 −λ)b−λa] ln

b

a
+ (2λ− 1)(b−a)

}
.

By virtue of the inequality in the right-hand side of (1.3), we obtain

∫ b
a

(t−a)λ(b− t)1−λ

t
d t >

∫ b
a

1

t

1
λ
t−a +

1−λ
b−t

d t

=
λ(1 −λ)(b−a)2

(2λ− 1)2(λb− (1 −λ)a)
ln

(
1

λ
− 1
)
−
ab(ln b− ln a)
λb− (1 −λ)a

+
b−a

2λ− 1
.

Substituting this into (1.1) yields

[λa + (1 −λ)b] −aλb1−λ >
sin(λπ)

π

{
λ(1 −λ)(b−a)2

(2λ− 1)2[λb− (1 −λ)a]
ln

(
1

λ
− 1
)

−
ab(ln b− ln a)
λb− (1 −λ)a

+
b−a

2λ− 1

}
→

1

π

b2 − 2ab(ln b− ln a) −a2

b−a

as λ → 1
2
. 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∫ b
a

(t−a)1−λ(b− t)λ

t2
d t <

∫ b
a

(1 −λ)(t−a) + λ(b− t)
t2

d t

= λ
b2 −a2

ab
+ (1 − 2λ) ln

b

a
+
a

b
− 1.

Substituting this into (1.2) yields

λa + (1 −λ)b
aλb1−λ

− 1 <
sin(λπ)

π

[
λ
b2 −a2

ab
+ (1 − 2λ) ln

b

a
+
a

b
− 1
]
.



134 F. QI

By virtue of the inequality in the right-hand side of (1.3), we obtain∫ b
a

(t−a)1−λ(b− t)λ

t2
d t >

∫ b
a

1

t2
1

1−λ
t−a +

λ
b−t

d t

=
(1 − 2λ)

[
λa2 − (1 −λ)b2

]
ln b

a
+ (b−a)

{
(2λ− 1)[λa− (1 −λ)b] + (1 −λ)λ(b−a) ln

(
1
λ
− 1
)}

(2λ− 1)[λa− (1 −λ)b]2

→
2(a + b)

b−a
ln
b

a
− 4

as λ → 1
2
. Substituting this into (1.2) yields

λa + (1 −λ)b
aλb1−λ

− 1 >
sin(λπ)

π

{
(1 −λ)b2 −λa2

[λa− (1 −λ)b]2
ln
b

a
+

(b−a)[λa− (1 −λ)b]
[λa− (1 −λ)b]2

+
(1 −λ)λ(b−a)2

(2λ− 1)[λa− (1 −λ)b]2
ln

(
1

λ
− 1
)}
→

2

π

[
(a + b)

b−a
ln
b

a
− 2
]

as λ → 1
2
. 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 = 1
2

and b > a > 0, the double inequality (1.4) can be written as

1

π

(
b−a

2
ln
b

a

)
>
a + b

2
−
√
ab >

2

π

(
a + b

2
−ab

ln b− ln a
b−a

)
> 0.

When λ = 1
2

and b > a > 0, the double inequality (1.5) can be written as

b−a
π

(
a + b

2ab
−

1

b

)
>

a + b

2
√
ab
− 1 >

2

π

[
(a + b)

b−a
ln
b

a
− 2
]
> 0.

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 ∈ C \ [−an,−a1].
Theorem 3.1 in [7] states that the principal branch of the weighted geometric mean

∏n
k=1(z + ak)

wk

has the integral representation

n∏
k=1

(z + ak)
wk =

n∑
k=1

wkak + z −
1

π

n−1∑
`=1

sin

[(∑̀
j=1

wj

)
π

]∫ a`+1
a`

n∏
k=1

|ak − t|wk
d t

t + z
. (3.1)

By the same arguments as in proofs of Theorems 1.1 and 1.2, we can derive from (3.1) lower and upper
bounds for the difference

∑n
k=1 wkak −

∏n
k=1 a

wk
k between the weighted arithmetic mean

∑n
k=1 wkak

and the geometric mean
∏n
k=1 a

wk
k .

Remark 3.4. In [4], it was obtained that, for ak < ak+1, z ∈ C \ [−an,−a1], and wk > 0 with∑n
k=1 wk = 1, the principal branch of the reciprocal of the weight geometric mean

∏n
k=1(z + ak)

wk

can be represented by

1∏n
k=1(z + ak)

wk
=

1

π

n−1∑
`=1

sin

(
π
∑̀
k=1

wk

)∫ a`+1
a`

1∏n
k=1 |t−ak|wk

1

t + z
d t. (3.2)

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

References

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Comput. Appl. Math. 311 (2017), 165–170.

[3] F. Qi and B.-N. Guo, The reciprocal of the weighted geometric mean is a Stieltjes function, Bol. Soc. Mat. Mex.
(2016). doi:10.1007/s40590-016-0151-5.

[4] F. Qi and B.-N. Guo, The reciprocal of the weighted geometric mean of many positive numbers is a Stieltjes function,

ResearchGate Working Paper (2016), doi:10.13140/RG.2.2.23822.36163.

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Acta Math. Sin. (Engl. Ser.) 30 (1) (2014), 61–68.

[8] F. Qi, X.-J. Zhang, and W.-H. Li, Lévy-Khintchine representation of the geometric mean of many positive numbers
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[9] F. Qi, X.-J. Zhang, and W.-H. Li, Lévy-Khintchine representations of the weighted geometric mean and the loga-
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(2016), doi:10.1007/s40590-016-0085-y.

1Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China

2College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia

Autonomous Region, 028043, China

3Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387,

China

∗Corresponding author: qifeng618@gmail.com, qifeng618@hotmail.com, qifeng618@qq.com


	1. Main results
	2. Proofs of Theorems 1.1 and 1.2
	3. Remarks
	References