In this article, we establish some new inequality chains for the ratio of certain bivariate means, and we present several sharp bounds for the arithmetic-geometric mean.
MSC:26E60, 26D07, 33E05.
Hinweise
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
1 Introduction
Let be the set of positive real numbers. Then a two-variable continuous function is said to be a mean on if the double inequality
holds for all .
Anzeige
The classical arithmetic-geometric mean of two positive real numbers a and b is defined as the common limit of sequences and , which are given by
where , , and for ,
(1.1)
The well-known Gauss identity shows that
for , where [1] is the complete elliptic integral of the first kind.
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Let with . Then the well-known Stolarsky mean [2] can be expressed as
(1.2)
Many bivariate means are the special case of the Stolarsky mean, for example, is the arithmetic mean, is the geometric mean, is the Heronian mean, is the logarithmic mean, is the identric (exponential) mean, the p-order arithmetic (power, Hölder) mean, is the p-order Heronian mean, is the p-order logarithmic mean, is the p-order identric (exponential) mean and is the one-parameter mean.
Another important family of means is the Gini means [3] defined by
(1.3)
it also contains many special means, for instance, is the arithmetic mean, is the power-exponential mean, is the p-order arithmetic (power, Hölder) mean, is the p-order power-exponential mean and is the Lehmer mean.
Recently, the inequalities for the bivariate means have been the subject of intensive research. In particular, the bounds for the arithmetic-geometric mean have attracted the attention of many mathematicians. It is well known that the double inequality
(1.4)
holds for all with . The first inequality of (1.4) was first proposed by Carlson and Vuorinen [4], it was proved in the literature [5‐8] by different methods. Vamanamurthy and Vuorinen [9] (also see [5, 6]) proved that for all with . The second inequality of (1.4) is due to Borwein and Borwein [10], and Yang [8] presented a simple proof by use of the ‘Comparison Lemma’ [[10], Lemma 2.1].
In [9] Vamanmurthy and Vuorinen presented the upper bounds for the arithmetic-geometric mean in terms of the arithmetic mean A, geometric mean G and logarithmic mean L as follows:
for all with .
In 1995, Sándor [5] proved that the double inequality
(1.5)
holds for all with , and it was improved by Alzer and Qiu [[11], Theorem 19] as
(1.6)
with the best possible parameters and .
Other inequalities involving can be found in the literature [12‐20].
The aim of this paper is to establish the new inequality chains for the ratio of certain bivariate means, and we present the sharp bounds for the arithmetic-geometric mean .
2 Lemmas
In order to establish our main results we need several lemmas, which we present in this section.
Lemma 7 is a consequence of the ‘Comparison Lemma’ in [[10], Lemma 2.1].
Lemma 7Let Φ be a bivariate mean such thatfor allwith . Then
for allwith .
3 Inequality chains for the ratio of means
In this section, we give some inequality chains for the ratio of certain bivariate means, which will be used to prove our main results in next section.
Proposition 1Letwith . Then we have
(3.1)
Proof The second inequality of (3.1) can be rewritten as
Therefore, it suffices to prove that the function
is strictly increasing in . Replacing x by and differentiating give
for .
Similarly, to prove the first inequality of (3.1), it suffices to prove that the function
is strictly increasing on . Replacing x by and differentiating yield
for , which completes the proof. □
Proposition 2Letwith . Then we have
(3.2)
whereand .
Proof By symmetry, without loss of generality, we assume that . Then from Lemma 5 and Proposition 1 we clearly see that the first and second inequalities of (3.2) hold. Next we prove the last inequality of (3.2). Let , then the last inequality of (3.2) can be rewritten as
It suffices to prove that the function
for .
Simple computations lead to
where
can be rewritten as
for . Therefore, for .
Thus we complete the proof. □
Proposition 3Letwithand . Then
Proof (i) From part one of Corollary 2 we see that
for .
Therefore, the first and second inequalities of Proposition 3 follow from the above inequalities and together with .
(ii)
For the third inequality of Proposition 3. From
we clearly see that it suffices to prove
Let . Then Corollary 1 leads to the conclusion that the function
is increasing in . Therefore, , that is,
(iii)
The fourth inequality of Proposition 3 can be written as , that is, . Let , then from Lemma 2 we know that is strictly decreasing with respect to .
(iv)
For the sixth, seventh, and eighth inequalities, let , then Lemma 2 leads to the conclusion that is strictly decreasing with respect to . Consequently,
which gives the desired results.
(v)
Finally, we prove the fifth inequality. It can be written as
Thus we need only to prove that the function
is strictly decreasing in . Let . Then
Differentiating yields
where
We clearly see that it is enough to prove for .
Making use of ‘product to sum’ and power series formulas we get
where
It is easy to verify that , . Next we show that for . To this end, we rewrite as
where
Due to for , it suffices to prove for , . Indeed,
therefore, ; ; .
This completes the proof. □
Proposition 4Letwith . Then forwe have
(3.3)
whereand .
Proof Without loss of generality, we assume that . Then the second inequality to the last inequality in (3.3) follows easily from Proposition 3 and Lemma 5.
Next we prove the first inequality of (3.3). Let . Then it equivalent to the inequality
Differentiating gives
where
Differentiating leads to
where
Making use of the power series we get
where
Clearly, , and for . Therefore, , is strictly increasing in , , , and for .
Thus the proof is finished. □
4 Sharp bounds for
In this section, we present several sharp bounds for the arithmetic-geometric mean .
Theorem 1 can be derived from Propositions 1-4 and Lemma 7.
Theorem 1Letwith . Then the inequalities
hold for .
Remark 1 We clearly see that the upper bound for is better than . Moreover, we have
Theorem 2The inequality
(4.1)
holds for allwithif and only if .
Proof Let and . Then (2.2) and the power series
lead to
Therefore, is the necessary condition for the inequality to hold for all with . The sufficiency follows easily from the function is strictly increasing and Theorem 1. □
Let with , and . Then we define by
(4.2)
Remark 2 From Theorem 1 and (1.6) we clearly see that the double inequality
holds for all with .
Moreover, making use of (2.1) and (2.2) we get
Therefore, and are the necessary conditions such that the double inequality
(4.3)
holds for all with .
Conjecture 1The double inequality
holds for allwithif and only ifand .
Theorem 3Let . Then the double inequality
(4.4)
holds for allwithif and only ifand .
Proof The sufficiency follows from the function is strictly decreasing in by Lemma 2 and Theorem 1.
Next we prove the necessity. It follows from (2.1) and (2.2) together with the power series
that
Therefore, and are the necessary conditions the double inequality (4.4) to hold for all with . □
Theorem 4Let . Then the inequality
(4.5)
holds for allwithif and only if .
Proof The sufficiency follows from the function is strictly decreasing in by Corollary 2(ii) and the inequality
in Theorem 1, and the necessity can be derived from the inequality
□
Making use of the similar methods, we can prove Theorems 5-7, we omit the proofs here.
Theorem 5Let and . Then the double inequality
holds for allwithif and only ifand .
Theorem 6The double inequality
holds for allwithif and only ifand .
Theorem 7The double inequality
holds for allwithif and only ifand .
Acknowledgements
This research was supported by the Natural Science Foundation of China under Grants 11371125, 11171307, and 61374086, and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.
Open Access
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https://creativecommons.org/licenses/by/2.0
), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.