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An Optimal Double Inequality between Power-Type Heron and Seiffert Means
Journal of Inequalities and Applications volume 2010, Article number: 146945 (2010)
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
.
1. Introduction
For , the power-type Heron mean
and the Seiffert mean
of two positive real numbers
and
are defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ1_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ3_HTML.gif)
For , the power mean
of order
of two positive numbers
and
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ4_HTML.gif)
The main properties for power mean are given in [21].
In [16], Jia and Cao presented the inequalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ5_HTML.gif)
for all with
,
, and
.
Sándor [22] proved that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ6_HTML.gif)
for all with
.
In [19], Seiffert established that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ7_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ9_HTML.gif)
Proof.
For , we clearly see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ10_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ11_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ12_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ13_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ14_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ15_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ16_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ17_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ18_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ19_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ20_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ21_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ22_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ23_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ24_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ25_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ26_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ27_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ28_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ29_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ30_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ31_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ32_HTML.gif)
From the expression of and Lemma 2.1, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ33_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ34_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ35_HTML.gif)
Then simple computations lead to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ36_HTML.gif)
where . Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ37_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ38_HTML.gif)
for .
Therefore, follows from (3.1)–(3.5).
Next, we prove that . Let
and
, then (1.1) and (1.2) lead to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ39_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ40_HTML.gif)
Then simple computations lead to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ41_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ42_HTML.gif)
where . Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ43_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ44_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ45_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ46_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ47_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ48_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ49_HTML.gif)
where .
Let , making use of Taylor extension, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F146945/MediaObjects/13660_2010_Article_2062_Equ50_HTML.gif)
Equations (3.15) and (3.17) together with inequality (3.16) imply that for any , there exist
and
such that
for
and
for
.
References
Wang M-K, Chu Y-M, Qiu Y-F: Some comparison inequalities for generalized Muirhead and identric means. Journal of Inequalities and Applications 2010, 2010:-10.
Long B-Y, Chu Y-M: Optimal inequalities for generalized logarithmic, arithmetic, and geometric means. Journal of Inequalities and Applications 2010, 2010:-10.
Long B-Y, Chu Y-M: Optimal power mean bounds for the weighted geometric mean of classical means. Journal of Inequalities and Applications 2010, 2010:-6.
Chu Y-M, Long B-Y: Best possible inequalities between generalized logarithmic mean and classical means. Abstract and Applied Analysis 2010, 2010:-13.
Chu Y-M, Qiu Y-F, Wang M-K, Wang G-D: The optimal convex combination bounds of arithmetic and harmonic means for the Seiffert's mean. Journal of Inequalities and Applications 2010, 2010:-7.
Long B, Xia W, Chu Y: An optimal inequality for power mean, geometric mean and harmonic mean. International Journal of Modern Mathematics 2010, 5(2):149–155.
Chu Y-M, Xia W-f: Two optimal double inequalities between power mean and logarithmic mean. Computers & Mathematics with Applications. 2010, 60(1):83–89. 10.1016/j.camwa.2010.04.032
Zhang X-M, Xi B-Y, Chu Y-M: A new method to prove and find analytic inequalities. Abstract and Applied Analysis 2010, 2010:-19.
Zhang X-M, Chu Y-M: A new method to study analytic inequalities. Journal of Inequalities and Applications 2010, 2010:-13.
Chu Y-M, Xia W-f: Inequalities for generalized logarithmic means. Journal of Inequalities and Applications 2009, 2009:-7.
Shi M-y, Chu Y-M, Jiang Y-p: Optimal inequalities among various means of two arguments. Abstract and Applied Analysis 2009, 2009:-10.
Chu Y-M, Xia W-f: Two sharp inequalities for power mean, geometric mean, and harmonic mean. Journal of Inequalities and Applications 2009, -6.
Chu Y, Xia W: Solution of an open problem for Schur convexity or concavity of the Gini mean values. Science in China A 2009, 52(10):2099–2106. 10.1007/s11425-009-0116-5
Chu Y, Zhang X: Necessary and sufficient conditions such that extended mean values are Schur-convex or Schur-concave. Journal of Mathematics of Kyoto University 2008, 48(1):229–238.
Chu Y, Zhang X, Wang G: The Schur geometrical convexity of the extended mean values. Journal of Convex Analysis 2008, 15(4):707–718.
Jia G, Cao J: A new upper bound of the logarithmic mean. Journal of Inequalities in Pure and Applied Mathematics 2003., 4(4, article 80):
Zhang Z-H, Lokesha V, Wu Y-d: The new bounds of the logarithmic mean. Advanced Studies in Contemporary Mathematics 2005, 11(2):185–191.
Zhang Z-H, Wu Y-d: The generalized Heron mean and its dual form. Applied Mathematics E-Notes 2005, 5: 16–23.
Seiffert J: Aufgabe
16. Die Wurzel 1995, 29: 221–222.
Hästö PA: A monotonicity property of ratios of symmetric homogeneous means. Journal of Inequalities in Pure and Applied Mathematics 2002., 3(5, article 71):
Bullen PS, Mitrinović DS, Vasić PM: Means and Their Inequalities. D. Reidel, , he Netherlands; 1988:xx+459.
Sándor J: A note on some inequalities for means. Archiv der Mathematik 1991, 56(5):471–473. 10.1007/BF01200091
Acknowledgments
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