An Optimal Double Inequality among the One-Parameter, Arithmetic and Geometric Means ()
1. Introduction
For
, the one-parameter mean
, arithmetic mean
and geometric mean
of two positive real numbers
and
are defined by
(1)
and
, respectively.
It is well-known that the one-parameter mean is continuous and strictly increasing with respect to
for fixed
with
. Many means are special cases of the one-parameter mean, for example:
is the arithmetic mean,
is the Heronian mean,
is the geometric mean, and
is the harmonic mean.
The one-parameter mean
and its inequalities have been studied intensively, see [1-6].
The purpose of this paper is to answer the question: for
, what are the greatest value
and the least value
such that the double inequality
holds for all
with
?
2. Main Result
The main result of this paper is the following theorem.
Theorem 2.1. Let
. Then for any
with
, we have 1)
for
2)
for
3)
for
.
The numbers
and
in 2) and 3) are optimal.
In order to prove Theorem 2.1, we need a preliminary lemma.
Lemma 2.1. For
, one has
(2)
Proof. Simple calculations lead to
(3)
(4)
(5)
(6)
(2) follows from (3)-(6).
Proof of Theorem 2.1. Without loss of generality we assume
and take
We first consider the case
. 1) follows from
![](https://www.scirp.org/html/1-1720057\a2101a7d-5c92-486f-9e26-2b989174cb9f.jpg)
From now on we assume
Let
then (1) leads to
(7)
where
![](https://www.scirp.org/html/1-1720057\f375655d-b857-4a0f-ba7a-34823fa4c26f.jpg)
Simple calculations lead to
(8)
![](https://www.scirp.org/html/1-1720057\7efe8ac6-34f2-4332-875c-c4ccbfbd63b1.jpg)
where
![](https://www.scirp.org/html/1-1720057\7735441f-228b-4bfb-9c3c-99ac09eb4a77.jpg)
(9)
![](https://www.scirp.org/html/1-1720057\b66fdf7f-1a71-4734-b4fe-f55a7d2a89e3.jpg)
(10)
(11)
where
![](https://www.scirp.org/html/1-1720057\67d08003-8b59-4223-b8f4-3bfc21f513c9.jpg)
(12)
(13)
where
(14)
(15)
(16)
We shall distinguish between two cases.
Case 1.
. The left-hand side inequality of 2)
for
follows from Lemma 2.1 because in this case
![](https://www.scirp.org/html/1-1720057\3b4e7afc-30de-4766-8889-2955683545b3.jpg)
for all
. In the sequel we assume
.
We clearly see from (16) that
![](https://www.scirp.org/html/1-1720057\7d97d775-6891-4a81-bbb5-4faefefa135d.jpg)
Thus
is strictly decreasing for
and strictly increasing for
. (2.14) yields
then
for
and
for
. The same reasoning applies to
and
as well, and noticing (13) and (12), one has
![](https://www.scirp.org/html/1-1720057\6e7e7fa6-1cdf-4dc3-ab5c-fdbbae157fa0.jpg)
This result together with (11) implies
![](https://www.scirp.org/html/1-1720057\0ee98738-8213-435a-aa66-4e8398d13859.jpg)
Thus
is strictly decreasing for
and strictly increasing for
The same reasoning applies to ![](https://www.scirp.org/html/1-1720057\0220eb67-ef21-4f46-812a-972cd72d22fb.jpg)
and
as well, and applying (8)-(10), we derive
![](https://www.scirp.org/html/1-1720057\cc470f15-9d27-42fe-b737-95411b8d8f35.jpg)
Since
for
and
for
, then we know from (7) that
![](https://www.scirp.org/html/1-1720057\eb1c8580-a46a-42aa-928c-7a994268a609.jpg)
This implies the left-hand side of 2) and the right-hand side of 3).
Case 2.
. From (14) we know that
![](https://www.scirp.org/html/1-1720057\b49a8181-eae8-4378-9b0a-bd96977b139f.jpg)
From (13) we know that
for
and
for
. This implies
is strictly decreasing for
and strictly increasing for
. From (12) we know
![](https://www.scirp.org/html/1-1720057\3b97b04f-8720-4a2b-8c4f-711614451dc2.jpg)
Therefore
![](https://www.scirp.org/html/1-1720057\abebb2fe-8dd0-4159-8291-f11dc2cd7c92.jpg)
(11) implies
has the same property as
thus
is strictly decreasing for
and strictly increasing for
. The same reasoning applies to
,
and
as well, and noticing (9) and (8), one has
![](https://www.scirp.org/html/1-1720057\2bc2a47d-bbf3-41f4-8fd4-dccc1f99de55.jpg)
which together with (7) implies
![](https://www.scirp.org/html/1-1720057\55911b08-ba0e-4232-a3e9-0a26b1f7c397.jpg)
This implies the right-hand side of 2) and the left-hand side of 3).
We are now in the position to prove the constants
and
are optimal.
For any
(positive or negative, with
sufficiently small) we consider the case
. (12)
implies
![](https://www.scirp.org/html/1-1720057\761129c3-5094-4b2f-8508-889b0c70140b.jpg)
By the continuity of
, there exists
such that
![](https://www.scirp.org/html/1-1720057\156ad935-a87b-4ba6-8699-b7cb03fab2e5.jpg)
By (11),
as the same property as
. The same reasoning applies to
,
,
and
as well, and noticing (10)-(8), we know
has the same property as
. By (7) one has
![](https://www.scirp.org/html/1-1720057\7ea58f38-9cd8-4666-99f2-f8e99cd5ec48.jpg)
This proves the optimality for
.
To prove the optimality for
in the right-hand side of 2) and the left-hand side of 3), we notice from
![](https://www.scirp.org/html/1-1720057\d9b81eed-385c-4cac-ae96-de7b74fe4464.jpg)
that there exists
such that
![](https://www.scirp.org/html/1-1720057\d141526e-6b1c-46e6-8fec-d4a0633ae612.jpg)
for
and
and
![](https://www.scirp.org/html/1-1720057\288005bc-7870-4eb7-85f0-4963abcc7ec8.jpg)
for
This ends the proof of Theorem 2.1.
3. Acknowledgements
This paper is supported by NSF of Hebei Province (A2011201011).