Abstract
For 
, the power-type Heron mean 
and the Seiffert mean 
of two positive real numbers 
and 
are defined by 
, 
; 
, 
and 
, 
; 
, 
, respectively. In this paper, we find the greatest value 
and the least value 
such that the double inequality 
holds for all 
with 
.
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1. Introduction
For 
, the power-type Heron mean 
and the Seiffert mean 
of two positive real numbers 
and 
are defined by
respectively.
Recently, the means of two variables have been the subject of intensive research [1–15]. In particular, many remarkable inequalities for 
and 
can be found in the literature [16–20].
It is well known that 
is continuous and strictly increasing with respect to 
for fixed 
with 
. Let 
, 
, 
, 
, and 
be the arithmetic, identric, logarithmic, geometric, and harmonic means of two positive numbers 
and 
with 
, respectively. Then
For 
, the power mean 
of order 
of two positive numbers 
and 
is defined by
The main properties for power mean are given in [21].
In [16], Jia and Cao presented the inequalities
for all 
with 
, 
, and 
.
Sándor [22] proved that
for all 
with 
.
In [19], Seiffert established that
for all 
with 
.
The purpose of this paper is to present the optimal upper and lower power-type Heron mean bounds for the Seiffert mean 
. Our main result is the following Theorem 1.1.
Theorem 1.1.
For all 
with 
, one has
and 
and 
are the best possible lower and upper power-type Heron mean bounds for the Seiffert mean 
, respectively.
2. Lemmas
In order to prove our main result, Theorem 1.1, we need two lemmas which we present in this section.
Lemma 2.1.
If 
and 
, then
Proof.
For 
, we clearly see that
Let
Then
and 
is strictly decreasing in 
because of 
for 
.
Therefore, Lemma 2.1 follows from (2.2)–(2.4) together with the monotonicity of 
.
Lemma 2.2.
If 
, 
, and 
, then there exists 
such that 
for 
and 
for 
.
Proof.
Let 
,
, 
, 
, 
, 
, and 
. Then elaborated computations lead to
From the expression of 
and Lemma 2.1, we get
From (2.24), we know that 
is strictly decreasing in 
. Then (2.22) implies that 
is strictly decreasing in 
.
From (2.20) and (2.21) together with the monotonicity of 
, we clearly see that there exists 
such that 
is strictly increasing in 
and strictly decreasing in 
.
Inequality (2.17) and (2.18) together with the piecewise monotonicity of 
imply that there exists 
such that 
is strictly increasing in 
and strictly decreasing in 
.
The piecewise monotonicity of 
together with (2.14) and (2.15) leads to the fact that there exists 
such that 
is strictly increasing in 
and strictly decreasing in 
.
From (2.11) and (2.12) together with the piecewise monotonicity of 
, we conclude that there exists 
such that 
is strictly increasing in 
and strictly decreasing in 
.
Equations (2.8) and (2.9) together with the piecewise monotonicity of 
imply that there exists 
such that 
is strictly increasing in 
and strictly decreasing in 
.
Therefore, Lemma 2.2 follows from (2.5) and (2.6) together with the piecewise monotonicity of 
.
3. Proof of Theorem 1.1
Proof of Theorem 1.1.
Without loss of generality, we assume that 
. We first prove that 
. Let 
, then from (1.1) and (1.2) we have
Let
Then simple computations lead to
where 
. Note that
where
for 
.
Therefore, 
follows from (3.1)–(3.5).
Next, we prove that 
. Let 
and 
, then (1.1) and (1.2) lead to
Let
Then simple computations lead to
where 
. Note that
where
From (3.12) and (3.13) together with Lemma 2.2, we clearly see that there exists 
such that 
is strictly increasing in 
and strictly decreasing in 
.
Equations (3.9)–(3.11) and the piecewise monotonicity of 
imply that there exists 
such that 
is strictly increasing in 
and strictly decreasing in 
. Then from (3.8) we get
for 
.
Therefore, 
follows from (3.6) and (3.7) together with (3.14).
At last, we prove that 
and 
are the best possible lower and upper power-type Heron mean bounds for the Seiffert mean 
, respectively.
For any 
and 
, from (1.1) and (1.2), one has
where 
.
Let 
, making use of Taylor extension, we get
Equations (3.15) and (3.17) together with inequality (3.16) imply that for any 
, there exist 
and 
such that 
for 
and 
for 
.
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This work was supported by the Natural Science Foundation of China under Grant no. 11071069, the Natural Science Foundation of Zhejiang Province under Grant no. Y7080106, and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant no. T200924.
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Chu, YM., Wang, MK. & Qiu, YF. An Optimal Double Inequality between Power-Type Heron and Seiffert Means. J Inequal Appl 2010, 146945 (2010). https://doi.org/10.1155/2010/146945
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DOI: https://doi.org/10.1155/2010/146945