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
From now on we assume 
Let 
then (1) leads to
(7)
where
Simple calculations lead to
(8)
where
(9)
(10)
(11)
where
(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
for all
. In the sequel we assume
.
We clearly see from (16) that
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
This result together with (11) implies
Thus 
is strictly decreasing for
and strictly increasing for 
The same reasoning applies to 
and 
as well, and applying (8)-(10), we derive
Since 
for 
and 
for
, then we know from (7) that
This implies the left-hand side of 2) and the right-hand side of 3).
Case 2.
. From (14) we know that
From (13) we know that 
for 
and 
for
. This implies 
is strictly decreasing for 
and strictly increasing for
. From (12) we know
Therefore
(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
which together with (7) implies
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
By the continuity of
, there exists 
such that
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
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
that there exists 
such that
for 
and 
and
for 
This ends the proof of Theorem 2.1.
3. Acknowledgements
This paper is supported by NSF of Hebei Province (A2011201011).