Abstract
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.
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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 .
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 ,
The well-known Gauss identity shows that
for , where [1] is the complete elliptic integral of the first kind.
Let with . Then the well-known Stolarsky mean [2] can be expressed as
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
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
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
holds for all with , and it was improved by Alzer and Qiu [[11], Theorem 19] as
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 1 ([[21], Corollary 1.1])
Let with . Then both and are strictly increasing (decreasing) with respect to () for fixed .
Lemma 2 ([[22], Theorem 5], [[23], Theorem 3.4])
Let with . Then the ratio of Stolarsky means is strictly increasing (decreasing) with respect to () for fixed .
Lemma 3 ([[24], Theorem 4.1])
Let with . Then the ratio of Stolarsky means is strictly log-concave (log-convex) with respect to () for fixed .
From Lemma 3, we have Corollary 1.
Corollary 1 Let , and with . Then the function
is strictly increasing in and strictly decreasing in .
Proof Let . Then
where ξ is between p and .
It follows from Lemma 3 that is strictly log-concave with respect to and strictly log-convex with respect to . Therefore, for and for . □
Lemma 4 ([[25], Corollary 3.1])
Let with . Then the function
is strictly decreasing (increasing) in and strictly increasing (decreasing) in for fixed , with .
Let , , respectively. Then Lemma 4 leads to the following.
Corollary 2 Let with . Then
-
(i)
the function
is strictly decreasing in and strictly increasing in ;
-
(ii)
the function
is strictly decreasing in and increasing in .
Lemma 5 Let with . Then .
Proof Simple computations lead to
□
Lemma 6 ([26])
Let . Then
Lemma 7 is a consequence of the ‘Comparison Lemma’ in [[10], Lemma 2.1].
Lemma 7 Let Φ be a bivariate mean such that for all with . Then
for all with .
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 1 Let with . Then we have
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 2 Let with . Then we have
where and .
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 3 Let with and . 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 4 Let with . Then for we have
where and .
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 1 Let with . Then the inequalities
hold for .
Remark 1 We clearly see that the upper bound for is better than . Moreover, we have
Theorem 2 The inequality
holds for all with if 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
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
holds for all with .
Conjecture 1 The double inequality
holds for all with if and only if and .
Theorem 3 Let . Then the double inequality
holds for all with if and only if and .
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 4 Let . Then the inequality
holds for all with if 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 5 Let and . Then the double inequality
holds for all with if and only if and .
Theorem 6 The double inequality
holds for all with if and only if and .
Theorem 7 The double inequality
holds for all with if and only if and .
References
Borwein JM, Borwein PB: Pi and the AGM. Wiley, New York; 1987.
Stolarsky KB: Generalizations of the logarithmic mean. Math. Mag. 1975, 48: 87–92. 10.2307/2689825
Gini C: Di una formula comprensiva delle medie. Metron 1938,13(2):3–22.
Carlson BC, Vuorinen M: Inequality of the AGM and the logarithmic mean. SIAM Rev. 1991,33(4):653–654.
Sándor J: On certain inequalities for means. J. Math. Anal. Appl. 1995,189(2):602–606. 10.1006/jmaa.1995.1038
Qi F, Sofo A: An alternative and united proof of a double inequality for bounding the arithmetic-geometric mean. Sci. Bull. “Politeh.” Univ. Buchar., Ser. A, Appl. Math. Phys. 2009,71(3):69–76.
Neuman E, Sándor J: On certain means of two arguments and their extensions. Int. J. Math. Math. Sci. 2003, 16: 981–993.
Yang Z-H: A new proof of inequalities for Gauss compound mean. Int. J. Math. Anal. 2010,4(21–24):1013–1018.
Vamanamurthy MK, Vuorinen M: Inequalities for means. J. Math. Anal. Appl. 1994,183(1):155–166. 10.1006/jmaa.1994.1137
Borwein JM, Borwein PB: Inequalities for compound mean iterations with logarithmic asymptotes. J. Math. Anal. Appl. 1993,177(2):572–582. 10.1006/jmaa.1993.1278
Alzer H, Qiu S-L: Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 2004,172(2):289–312. 10.1016/j.cam.2004.02.009
Qiu S-L, Shen J-M: On two problems concerning means. J. Hangzhou Inst. Electron. Eng. 1997,17(3):1–7. (in Chinese)
Sándor J: On certain inequalities for means II. J. Math. Anal. Appl. 1996,199(2):629–635. 10.1006/jmaa.1996.0165
Sándor J: On certain inequalities for means III. Arch. Math. 2001,76(1):34–40. 10.1007/s000130050539
Toader G: Some mean values related to the arithmetic-geometric mean. J. Math. Anal. Appl. 1998,218(2):358–368. 10.1006/jmaa.1997.5766
Chung S-Y: Functional means and harmonic functional means. Bull. Aust. Math. Soc. 1998,57(2):207–220. 10.1017/S0004972700031609
Bracken P: An arithmetic-geometric mean inequality. Expo. Math. 2001,19(3):273–279. 10.1016/S0723-0869(01)80006-2
Liu Z: Compounding of Stolarsky means. Soochow J. Math. 2004,30(2):149–163.
Liu Z: Remarks on arithmetic-geometric mean and geometric-harmonic mean. J. Anshan Univ. Sci. Technol. 2007,30(3):230–235. (in Chinese)
Neuman E: Inequalities for weighted sums of powers and their applications. Math. Inequal. Appl. 2012,15(4):995–1005.
Yang Z-H: The log-convexity of another class of one-parameter means and its applications. Bull. Korean Math. Soc. 2012,49(1):33–47. 10.4134/BKMS.2012.49.1.033
Losonczi L: Ratio of Stolarsky means: monotonicity and comparison. Publ. Math. (Debr.) 2009,75(1–2):221–238.
Yang Z-H: Some monotonicity results for the ratio of two-parameter symmetric homogeneous functions. Int. J. Math. Math. Sci. 2009., 2009: Article ID 591382
Yang Z-H: Log-convexity of ratio of the two-parameter symmetric homogeneous functions and an application. J. Inequal. Spec. Funct. 2010,1(1):16–29.
Yang Z-H: The monotonicity results for the ratio of certain mixed means and their applications. Int. J. Math. Math. Sci. 2012., 2012: Article ID 540710
Peetre J: Generalizing the arithmetic geometric mean - a hapless computer experiment. Int. J. Math. Math. Sci. 1989,12(2):235–245. 10.1155/S016117128900027X
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.
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Yang, ZH., Song, YQ. & Chu, YM. Sharp bounds for the arithmetic-geometric mean. J Inequal Appl 2014, 192 (2014). https://doi.org/10.1186/1029-242X-2014-192
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DOI: https://doi.org/10.1186/1029-242X-2014-192