New sharp bounds for logarithmic mean and identric mean
© Yang; licensee Springer 2013
Received: 9 November 2012
Accepted: 22 February 2013
Published: 20 March 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 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.
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.
where . And then, due to Lemma 1, Theorem 1 and Theorem 2 can be restated as equivalent ones, respectively.
holds for all if and only if and the function is decreasing on . Inequality (1.16) is reversed if and only if .
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.
which proves the lemma. □
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 .
Now we are ready to prove desired results.
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.
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).
which yields .
for all .
which leads to .
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.
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 .
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).
which leads to .
for all .
Solving the inequalities yields .
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.
holds for with , where the constants and are the best constants.
By Theorem 2 and (2.3), we obtain the following.
holds for with , where and are the best constants.
where with .
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.
hold for with , where .
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.
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 .
where is the best constant.
which is equivalent to (4.10).
Thus the proof is completed. □
5 Comparison of some lower bounds for logarithmic mean
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 .
hold for all with , where and cannot be improved.
Clearly, if we prove that for all , if and if , then inequalities (5.1) are valid.
which together with (5.3) yields under the inductive assumption for . By mathematical induction, it is acquired that for all .
for all .
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 .
which completes the proof. □
Using Lemmas 7-9, we can easily prove the following.
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 ().
holds for all with if . While and are not comparable if .
which proves inequality (5.6).
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.
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.
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