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Optimal Inequalities for Generalized Logarithmic, Arithmetic, and Geometric Means
Journal of Inequalities and Applications volume 2010, Article number: 806825 (2010)
Abstract
For , the generalized logarithmic mean , arithmetic mean , and geometric mean of two positive numbers and are defined by , for , , for , , and , , for , and , , for , and , , and , respectively. In this paper, we find the greatest value (or least value , resp.) such that the inequality (or , resp.) holds for (or , resp.) and all with .
1. Introduction
For , the generalized logarithmic mean and power mean with parameter of two positive numbers and are defined by
and
respectively. It is well known that both means are continuous and increasing with respect to for fixed and . Recently, both means have been the subject of intensive research. In particular, many remarkable inequalities involving and can be found in the literature [1–9]. Let
, and be the arithmetic, identric, logarithmic, geometric, and harmonic means of two positive numbers and , respectively. Then
for all .
In [10], Carlson proved that
for all with .
The following inequality is due to Sándor [11, 12]:
In [13], Lin established the following results: (1) implies that for all with ; (2) implies that for all with ; (3) implies that there exist such that ; (4) implies that there exist such that . Hence the question was answered: what are the least value and the greatest value such that the inequality holds for all with .
Pittenger [14] established that
for all , where
Here, and are sharp and inequality (1.7) becomes equality if and only if or or . The case reduces to Lin's results [13]. Other generalizations of Lin's results were given by Imoru [15].
Recently, some monotonicity results of the ratio between generalized logarithmic means were established in [16–18].
The aim of this paper is to prove the following Theorem 1.1.
Theorem 1.1.
Let and with , then
(1) for ;
(2) for , and for , moreover, in each case, the bound for the sum is optimal.
2. Proof of Theorem 1.1
In order to prove our Theorem 1.1 we need a lemma, which we present in this section.
Lemma 2.1.
For and one has
(1)If , then for ;
(2)If , then for , for , and for .
Proof.

(1)
If , then we clearly see that
(2.1)
for .
If , then
for .
Therefore, Lemma 2.1() follows from (2.1) and (2.2).

(2)
If , then
(2.3)
Therefore, Lemma 2.1(2) follows from (2.3).
Proof of Theorem 1.1.
Proof.
() If , then (1.1) leads to
() We divide the proof into two cases.
Case 1.
or . From inequalities (1.5) and (1.6) we clearly see that
for , and
for .
Case 2.
. Without loss of generality, we assume that . Let , then (1.1) leads to
Let , then simple computations yield
where
Note that
where is defined as in Lemma 2.1.
We divide the proof into five subcases.
Subcase 2 A.
. From (2.18) and Lemma 2.1() we clearly see that for and for , then we know that is strictly decreasing in and strictly increasing in . Now from the monotonicity of and (2.17) together with the fact that we clearly see that for , then from (2.7)–(2.15) and for we get for .
Subcase 2 B.
. Then (2.18) and Lemma 2.1(1) lead to
for .
From (2.7)–(2.17) and (2.19) together with the fact that for we know that for .
Subcase 2 C.
. Then (2.18) and Lemma 2.1(1) imply that
for .
From (2.7)–(2.17), (2.20) and for we know that for .
Subcase 2 D.
. Then (2.19) again yields, and for follows from (2.7)–(2.17) and (2.19) together with .
Subcase 2 E.
. Then (2.20) is also true, and for follows from (2.7)–(2.17), (2.20) and the fact that .
Next, we prove that the bound for the sum is optimal in each case. The proof is divided into six cases.
Case 1.
. For any and , then (1.1) leads to
where
Let making use of Taylor expansion, one has
Equations (2.21) and (2.22) imply that for any , there exists , such that for any and .
Case 2.
. For any and , from (1.1) we have
where
Let making use of Taylor expansion, one has
Equations (2.23) and (2.24) imply that for any , there exists , such that for and .
Case 3.
. For and , we get
where
Let making use of Taylor expansion, one has
Equations (2.25) and (2.26) imply that for any and any , there exists , such that for .
Case 4.
. For any and , we get
where
Let using Taylor expansion we have
Equations (2.27) and (2.28) show that for any and any , there exists , such that for .
Case 5.
. For any and , we have
where
Let making use of Taylor expansion we get
Equations (2.29) and (2.30) imply that for any and any , there exists , such that for .
Case 6.
. For any and , we get
where
Let , using Taylor expansion we have
From (2.31) and (2.32) we know that for any and any , there exists , such that for .
At last, we propose two open problems as follows.
Open Problem 1
What is the least value such that the inequality
holds for and all with ?
Open Problem 2
What is the greatest value such that the inequality
holds for and all with ?
References
Alzer H: Ungleichungen für Mittelwerte. Archiv der Mathematik 1986, 47(5):422–426. 10.1007/BF01189983
Alzer H, Qiu SL: Inequalities for means in two variables. Archiv der Mathematik 2003, 80(2):201–215. 10.1007/s0001300304562
Burk F: The geometric, logarithmic and arithmetic mean inequality. The American Mathematical Monthly 1987, 94(6):527–528. 10.2307/2322844
Janous W: A note on generalized Heronian means. Mathematical Inequalities & Applications 2001, 4(3):369–375.
Leach EB, Sholander MC: Extended mean values. II. Journal of Mathematical Analysis and Applications 1983, 92(1):207–223. 10.1016/0022247X(83)902809
Sándor J: On certain inequalities for means. Journal of Mathematical Analysis and Applications 1995, 189(2):602–606. 10.1006/jmaa.1995.1038
Sándor J: On certain inequalities for means. II. Journal of Mathematical Analysis and Applications 1996, 199(2):629–635. 10.1006/jmaa.1996.0165
Sándor J: On certain inequalities for means. III. Archiv der Mathematik 2001, 76(1):34–40. 10.1007/s000130050539
Shi MY, Chu YM, Jiang YP: Optimal inequalities among various means of two arguments. Abstract and Applied Analysis 2009, 2009:10.
Carlson BC: The logarithmic mean. The American Mathematical Monthly 1972, 79: 615–618. 10.2307/2317088
Sándor J: On the identric and logarithmic means. Aequationes Mathematicae 1990, 40(2–3):261–270.
Sándor J: A note on some inequalities for means. Archiv der Mathematik 1991, 56(5):471–473. 10.1007/BF01200091
Lin TP: The power mean and the logarithmic mean. The American Mathematical Monthly 1974, 81: 879–883. 10.2307/2319447
Pittenger AO: Inequalities between arithmetic and logarithmic means. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika 1981, (678–715):15–18.
Imoru CO: The power mean and the logarithmic mean. International Journal of Mathematics and Mathematical Sciences 1982, 5(2):337–343. 10.1155/S0161171282000313
Chen ChP: The monotonicity of the ratio between generalized logarithmic means. Journal of Mathematical Analysis and Applications 2008, 345(1):86–89. 10.1016/j.jmaa.2008.03.071
Li X, Chen ChP, Qi F: Monotonicity result for generalized logarithmic means. Tamkang Journal of Mathematics 2007, 38(2):177–181.
Qi F, Chen ShX, Chen ChP: Monotonicity of ratio between the generalized logarithmic means. Mathematical Inequalities & Applications 2007, 10(3):559–564.
Acknowledgments
The authors wish to thank the anonymous referee for their very careful reading of the manuscript and fruitful comments and suggestions. This research is partly supported by N S Foundation of China under Grant 60850005, and N S Foundation of Zhejiang Province under Grants D7080080 and Y607128.
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Long, BY., Chu, YM. Optimal Inequalities for Generalized Logarithmic, Arithmetic, and Geometric Means. J Inequal Appl 2010, 806825 (2010). https://doi.org/10.1155/2010/806825
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DOI: https://doi.org/10.1155/2010/806825