Research Article
Open Access
The Optimal Convex Combination Bounds for Seiffert's Mean
- Hong Liu1Email author and
- Xiang-Ju Meng2
Journal of Inequalities and Applications20112011:686834
https://doi.org/10.1155/2011/686834
© H. Liu and X.-J. Meng. 2011
Received: 28 November 2010
Accepted: 28 February 2011
Published: 13 March 2011
Abstract
We derive some optimal convex combination bounds related to Seiffert's mean. We find the greatest values
,
and the least values
,
such that the double inequalities
and
hold for all
with
. Here,
,
,
, and
denote the contraharmonic, geometric, harmonic, and Seiffert's means of two positive numbers
and
, respectively.
1. Introduction
Recently, the inequalities for means have been the subject of intensive research. In particular, many remarkable inequalities for
can be found in the literature [2–6]. Seiffert's mean
can be rewritten as (see [5, equation (2.4)])
can be found in the literature [2–6]. Seiffert's mean
can be rewritten as (see [5, equation (2.4)])
(1.2)
Let
, and
be the contraharmonic, arithmetic, geometric and harmonic means of two positive real numbers
and
with
. Then
, and
be the contraharmonic, arithmetic, geometric and harmonic means of two positive real numbers
and
with
. Then
(1.3)
In [7], Seiffert proved that
(1.4)
for all
with
.
In [8], the authors found the greatest value
and the least value
such that the double inequality
and the least value
such that the double inequality
(1.5)
holds for all
with
.
The purpose of the present paper is to find the greatest values
and the least values
such that the double inequalities
and the least values
such that the double inequalities
(1.6)
hold for all
with
.
2. Main Results
Firstly, we present the optimal convex combination bounds of contraharmonic and geometric means for Seiffert's mean as follows.
Theorem 2.1.
The double inequality
holds for all
with
if and only if
and
.
Proof.
Firstly, we prove that
(2.1)
for all
with
.
Without loss of generality, we assume that
. Let
and
. Then (1.1) leads to
. Let
and
. Then (1.1) leads to
(2.2)
where
(2.3)
Simple computations lead to
(2.4)
where
(2.5)
We divide the proof into two cases.
Case 1 (
).
In this case,
(2.6)
Therefore, the second inequality in (2.1) follows from (2.2)–(2.6). Notice that in this case, the second equality in (2.4) becomes
(2.7)
Case 2 (
).
From (2.5), we have that
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
From (2.17) and (2.18), we clearly see that
for
; hence
is strictly increasing in
, which together with (2.16) implies that there exists
such that
for
and
for
; and hence
is strictly decreasing in
and strictly increasing for
. From (2.14) and the monotonicity of
, there exists
such that
for
and
for
; hence
is strictly decreasing in
and strictly increasing for
. As this goes on, there exists
such that
is strictly decreasing in
and strictly increasing in
. Note that if
, then the second equality in (2.4) becomes
for
; hence
is strictly increasing in
, which together with (2.16) implies that there exists
such that
for
and
for
; and hence
is strictly decreasing in
and strictly increasing for
. From (2.14) and the monotonicity of
, there exists
such that
for
and
for
; hence
is strictly decreasing in
and strictly increasing for
. As this goes on, there exists
such that
is strictly decreasing in
and strictly increasing in
. Note that if
, then the second equality in (2.4) becomes
(2.19)
Thus
for all
. Therefore, the first inequality in (2.1) follows from (2.2) and (2.3).
Secondly, we prove that
is the best possible lower convex combination bound of the contraharmonic and geometric means for Seiffert's mean.
If
, then (2.5) (with
in place of
) leads to
, then (2.5) (with
in place of
) leads to
(2.20)
From this result and the continuity of
we clearly see that there exists
such that
for
. Then the last equality in (2.4) implies that
for
. Thus
is decreasing for
. Due to (2.4),
for
, which is equivalent to, by (2.2),
we clearly see that there exists
such that
for
. Then the last equality in (2.4) implies that
for
. Thus
is decreasing for
. Due to (2.4),
for
, which is equivalent to, by (2.2),
(2.21)
for
.
Finally, we prove that
is the best possible upper convex combination bound of the contraharmonic and geometric means for Seiffert's mean.
If
, then from (1.1) one has
, then from (1.1) one has
(2.22)
Inequality (2.22) implies that for any
there exists
such that
there exists
such that
(2.23)
for
.
Secondly, we present the optimal convex combination bounds of the contraharmonic and harmonic means for Seiffert's mean as follows.
Theorem 2.2.
The double inequality
holds for all
with
if and only if
and
.
Proof.
Firstly, we prove that
(2.24)
for all
with
.
Without loss of generality, we assume that
. Let
and
. Then (1.1) leads to
. Let
and
. Then (1.1) leads to
(2.25)
where
(2.26)
Simple computations lead to
(2.27)
where
(2.28)
We divide the proof into two cases.
Case 1 (
).
In this case,
(2.29)
Therefore, the first inequality in (2.24) follows from (2.25)–(2.29). Notice that in this case, the second equality in (2.27) becomes
(2.30)
Case 2 (
).
From (2.28) we have that
(2.31)
(2.32)
(2.33)
(2.34)
(2.35)
(2.36)
(2.37)
(2.38)
(2.39)
(2.40)
(2.41)
From (2.40) and (2.41) we clearly see that
for
; hence
is strictly decreasing in
, which together with (2.39) implies that there exists
such that
for
and
for
, and hence
is strictly increasing in
and strictly decreasing for
. From (2.37) and the monotonicity of
, there exists
such that
for
and
for
; hence
is strictly increasing in
and strictly decreasing for
. As this goes on, there exists
such that
is strictly increasing in
and strictly decreasing in
. Notice that if
, then the second equality in (2.27) becomes
for
; hence
is strictly decreasing in
, which together with (2.39) implies that there exists
such that
for
and
for
, and hence
is strictly increasing in
and strictly decreasing for
. From (2.37) and the monotonicity of
, there exists
such that
for
and
for
; hence
is strictly increasing in
and strictly decreasing for
. As this goes on, there exists
such that
is strictly increasing in
and strictly decreasing in
. Notice that if
, then the second equality in (2.27) becomes
(2.42)
Thus
for all
. Therefore, the second inequality in (2.24) follows from (2.25) and (2.26).
Secondly, we prove that
is the best possible upper convex combination bound of the contraharmonic and harmonic means for Seiffert's mean.
If
, then (2.28) (with
in place of
) leads to
, then (2.28) (with
in place of
) leads to
(2.43)
From this result and the continuity of
we clearly see that there exists
such that
for
. Then the last equality in (2.27) implies that
for
. Thus
is increasing for
. Due to (2.27),
for
, which is equivalent to, by (2.25),
we clearly see that there exists
such that
for
. Then the last equality in (2.27) implies that
for
. Thus
is increasing for
. Due to (2.27),
for
, which is equivalent to, by (2.25),
(2.44)
for
.
Finally, we prove that
is the best possible lower convex combination bound of the contraharmonic and harmonic means for Seiffert's mean.
If
, then from (1.1) one has
, then from (1.1) one has
(2.45)
Inequality (2.45) implies that for any
there exists
such that
there exists
such that
(2.46)
for
.
Declarations
Acknowledgments
The authors wish to thank the anonymous referees for their very careful reading of the paper and fruitful comments and suggestions. This research is partly supported by N S Foundation of Hebei Province (Grant A2011201011), and the Youth Foundation of Hebei University (Grant 2010Q24).
Authors’ Affiliations
(1)
College of Mathematics and Computer Science, Hebei University
(2)
Department of Mathematics, Baoding College
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© H. Liu and X.-J. Meng. 2011
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
