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New sharp bounds for logarithmic mean and identric mean
Journal of Inequalities and Applications volume 2013, Article number: 116 (2013)
For with , let , , , , denote the logarithmic mean, identric mean, arithmetic mean, geometric mean and r-order power mean, respectively. We find the best constant such that the inequalities
hold, respectively. From them some new inequalities for means are derived. Lastly, our new lower bound for the logarithmic mean is compared with several known ones, which shows that our results are superior to others.
MSC: 26D07, 26E60, 05A15, 15A18.
The logarithmic and identric means of two positive real numbers x and y with are defined by
respectively. The power mean of order r of the positive real numbers x and y is defined by
The main properties of these means are given in . In particular, the function () is continuous and strictly increasing on ℝ. As special cases, the arithmetic mean and geometric mean are and , respectively.
In 1974 Lin  obtained an important refinement of the above inequalities:
In , the authors present a very nice double inequality, that is,
Using a new method, Wang and Wang  proved that
holds for . Chen and Wang  pointed out this inequality is true for all real numbers p. Only when , however, the inequality would be true. In 2009, another better lower bound for L was given by Zhu , that is,
The following lower bound for L in terms of I and G is due to Alzer :
Very recently, Yang  showed that
The aim of this paper is to find the best such that the inequalities
It is easy to check that both the functions
are even on , and therefore we assume that in what follows.
Our main results are stated as follows.
Theorem 1 Let . Then inequalities (1.12) hold for all with if and only if and , and the function is decreasing on .
Theorem 2 Let . Then inequalities (1.13) hold for all with if and only if and , and the function is decreasing on .
We will prove two theorems above by hyperbolic function theory. For this end, we need the following lemma, which tells us an inequality for bivariate homogeneous means can be equivalently changed into the form of hyperbolic functions.
Lemma 1 Let be a homogeneous mean of positive arguments x and y. Then
By symmetry, we assume that . Then we have
where . And then, due to Lemma 1, Theorem 1 and Theorem 2 can be restated as equivalent ones, respectively.
Theorem 1′ Let . Then the inequality
holds for all if and only if and the function is decreasing on . Inequality (1.16) is reversed if and only if .
Theorem 2′ Let . Then the inequality
holds for all if and only if and the function is decreasing on . Inequality (1.17) is reversed if and only if .
Therefore, we will prove Theorem 1′ and Theorem 2′ instead of Theorem 1 and Theorem 2 in the sequel.
2 Proof of Theorem 1′
In order to prove Theorem 1′, we first give the following lemmas.
Lemma 2 For , let the function be defined by
Then U is decreasing on with
which implies that V is decreasing on , and so . Therefore, , that is to say, U is decreasing on . And, by L’Hospital’s rule, we have
which proves the lemma. □
Remark 1 From Lemmas 1 and 2 it follows that the function is decreasing on , and
so are the functions defined by (1.14) and (1.15), and
Lemma 3 Let and let be the function defined by
Then we have
Proof Using L’Hospital’s rule gives (2.5). To obtain (2.6), we write as
from which (2.6) easily follows.
This lemma is proved. □
Lemma 4 For , let f be defined by (2.4). Then f is increasing if and decreasing if .
Using ‘product into sum’ formula for hyperbolic functions gives
and expanding in power series gives
It is easy to check that
Now we are ready to prove desired results.
(i) We first prove that f is increasing if . To this end, by (2.7) in combination with (2.8) and (2.9), it suffices to show that for . We easily check that and
Assume that for . From (2.12) it is easy to see that
which together with (2.11) yields , and from (2.10) it is derived that . By mathematical induction, we conclude that for , which proves part one of this lemma.
(ii) Next we show that f is decreasing if . Likewise, it needs to be shown that for . As mentioned previously, but
Suppose that for . We have
which leads to , and from (2.10) we have for . By mathematical induction, it is obtained that for .
This completes the proof. □
Now we prove Theorem 1′.
Proof of Theorem 1′ It is clear that (1.16) (or its reverse inequality) is equivalent to (or <0), where is defined by (2.4).
(i) We first show that for all if and only if . If for all , then by (2.5) and (2.6) we have
which yields .
Conversely, if , then by Lemma 4 f is increasing on , hence
for all .
(ii) Next we prove that for all if and only if . If for all , then by (2.5) and (2.6) we have
which leads to .
Conversely, if , then from the monotonicity of f by Lemma 4 we conclude that
for all .
(iii) Lastly, from Lemma 2 we easily conclude that the function is decreasing on .
Thus the proof is accomplished. □
3 Proof of Theorem 2′
The following lemmas are useful.
Lemma 5 Let and let be the function defined by
Then we have
Proof Since as , using L’Hospital’s rule yields (3.2). To obtain (3.3), we have to change as follows:
from which (3.3) follows.
Thus the lemma is proved. □
Lemma 6 For , let the function g be defined by (3.1). Then g is increasing if and decreasing if .
Using ‘product into sum’ formula for hyperbolic functions and expanding in power series give
We find that
We claim that
Indeed, applying the binomial expansion gives
Hence, if , then we get
that is, (3.9) holds. If , then
for , which in combination with
leads to (3.10).
Now we are in a position to prove our results.
(i) We first prove that g is increasing if . For this end, it is enough to show that by (3.5) and (3.6). Indeed, we have and . Suppose that for . From (3.8) and (3.9) we have , which proves part one of this lemma by mathematical induction.
(ii) Next we prove that g is decreasing if . It suffices to prove that . We have seen that , but . Using (3.8) and (3.10), we conclude that if for . By mathematical induction, part two of this lemma is proved.
Thus the proof ends. □
Based on the above lemmas, Theorem 2′ can be easily proved.
Proof of Theorem 2′ It is clear that (1.17) (or its reverse inequality) is equivalent to (or <0), where is defined by (3.1).
(i) We first prove that for all if and only if . If for all , then by (3.2) and (3.3) we have
which leads to .
Conversely, if , then by Lemma 6 we get
for all .
(ii) Next we show that for all if and only if . If for all , then by (3.2) and (3.3) we obtain
Solving the inequalities yields .
Conversely, if , then by the monotonicity of g we have
for all .
(iii) Lastly, due to Lemma 2, the function is clearly decreasing on .
This proves the proof. □
Using Theorem 1 and (2.2), the following corollaries are immediate.
Corollary 1 We have
holds for with , where the constants and are the best constants.
By Theorem 2 and (2.3), we obtain the following.
Corollary 2 We have
holds for with , where and are the best constants.
Remark 2 Neuman  has derived some bounds for certain differences of bivariate means, one of which is as follows:
While (4.1) and (4.2) contain some new bounds for certain ratios of bivariate means, for example,
where with .
Making use of identity for means
Employing Theorem 1, Theorem 2 and (2.2), we can prove an interesting chain of inequalities involving the logarithmic mean, identric mean, power mean and geometric mean.
Corollary 3 Let , , . Then the inequalities
hold for with , where .
Proof By Remark 1, we see that the function is decreasing on , and by (2.2) it is deduced that
if and .
The second and third inequalities are equivalent to (1.17), which hold if and only if and by Theorem 2, respectively.
If , then by Theorem 1 the fourth and fifth inequalities hold.
With , , , by Theorem 2, we have
and for , that is, ,
Squaring both sides of (4.8) and (4.9) yields the sixth and seventh inequality, respectively.
The proof is finished. □
From Lemma 6 with (3.4) another known interesting inequality can be reobtained. It should be noted that the second inequality in (4.10) first appeared in  and was reproved by Neuman and Sándor .
Corollary 4 For with , we have
where is the best constant.
Proof From (3.4) it is obtained that
if . And since the function is increasing on by Lemma 6, we have
which is equivalent to (4.10).
Thus the proof is completed. □
5 Comparison of some lower bounds for logarithmic mean
As mentioned in the first section of this paper, there are many lower bounds for the logarithmic mean L such as
and so on, some of which have been proved to be comparable and others remain to be compared further. As applications of our main results, we will discuss them in this section. To this end, we first give a lemma.
Lemma 7 ([, Conclusion 1])
The function is strictly log-concave on .
Now we compare with .
Lemma 8 Let . Then the inequalities
hold for all with , where and cannot be improved.
Proof By Lemma 1, to prove (5.1), it suffices to show that
For , we define
Differentiation and expanding in power series lead to
We easily establish a recursive relation for the sequence :
Clearly, if we prove that for all , if and if , then inequalities (5.1) are valid.
Now we show that for all , if . In fact, it is easy to verify that , and due to ,
which together with (5.3) yields under the inductive assumption for . By mathematical induction, it is acquired that for all .
We next prove that for all , if . It is not difficult to get
for all .
Lastly, we prove and cannot be improved. Indeed, if for all , then we have
which yields . On the other hand, by Lemma 2 the function is decreasing on . Therefore, is the largest constant such that for all .
In the same way, we can prove is the smallest constant such that for all .
This completes the proof. □
Next let us compare and .
Lemma 9 For with , we have
Proof Suppose that and let . Then inequality (5.4) can be equivalently changed into
where . Simplifying yields
which completes the proof. □
Using Lemmas 7-9, we can easily prove the following.
Proposition 1 For with and , , we have
Proof The first, second and third inequalities follow from Theorem 1 and Lemma 8. Since , by Theorem 1 the fourth and fifth ones follow. The sixth and seventh ones are obtained from the second and third ones of (4.7).
By the known inequality (see [, (5.13)]), we easily get the eighth one.
Lemma 9 shows that the ninth one holds.
The last one is equivalent to , which easily follows from Lemma 7.
This completes the proof. □
Lastly, we compare with ().
Proposition 2 Let and . Then
holds for all with if . While and are not comparable if .
Proof (i) By a simple equivalent transformation, inequality (5.6) can be changed into
where , . If , then , with . By Lemma 7, it is derived that
and using the basic inequality for and the monotonicity of the function , we get
which proves inequality (5.6).
(ii) We define
By Lemma 1, to show that and are not comparable if , we need to illustrate is not a constant. In fact, utilizing L’Hospital’s rule, we have
which implies that there are numbers such that when and when . Consequently, and are not comparable on if , which is the desired result.
Thus the proof is finished. □
Remark 3 From the above two propositions, as far as the lower bounds for the logarithmic mean are concerned, our new lower bound seems to be superior to most known ones.
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The author would like to thank Mr. Pi for her help. The author also wishes to thank the reviewer(s) who gave some important and valuable advises.
The author declares that they have no competing interests.
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Yang, ZH. New sharp bounds for logarithmic mean and identric mean. J Inequal Appl 2013, 116 (2013). https://doi.org/10.1186/1029-242X-2013-116
- logarithmic mean
- identric mean
- power mean