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József Sándor, "On Huygens' Inequalities and the Theory of Means", International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 597490, 9 pages, 2012. https://doi.org/10.1155/2012/597490
On Huygens' Inequalities and the Theory of Means
By using the theory of means, various refinements of Huygens' trigonometric and hyperbolic inequalities will be proved. New Huygens' type inequalities will be provided, too.
Let be two positive real numbers. The logarithmic and identric means of and are defined by respectively. Seiffert's mean is defined by
Let denote the arithmetic, geometric, and harmonic means of and , respectively. These means have been also in the focus of many research papers in the last decades. For a survey of results, see, for example, [4–6]. In what follows, we will assume .
Now, by remarking that letting , , where , in , we find that so Huygens' inequality (1.1) may be written also as Here denotes the harmonic mean of the numbers :
On the other hand, by letting , in , we find that so Huygens' hyperbolic inequality (1.2) may be written also as
2. First Improvements
Suppose , .
Theorem 2.1. One has
Proof. The inequalities and have been proved in paper  (see Corollary 3.2). In fact, stronger relations are valid, as we will see in what follows.
Now, the interesting fact is that the second inequality of (2.1), that is, becomes, after elementary transformations, exactly inequality (1.10), while the second inequality of (2.2), that is, becomes inequality (1.7).
Theorem 2.2. One has the inequalities:
Proof. The first inequality of (2.3) is proved in , while the first inequality of (2.8) is a well-known inequality due to Leach and Sholander  (see  for many related references). The second inequalities of (2.3) and (2.4) are immediate consequences of the arithmetic-geometric inequality applied for , , and , , , respectively.
Remark 2.3. By (2.3) and (1.6), we can deduce the following improvement of the Huygens' inequality (1.1):
From (2.1) and (1.6), we get
Similarly, by (2.4) and (1.9), we get
From (2.2) and (1.9), we get
Here, , .
We note that the first inequality of (2.5) has been discovered by Adamović and Mitrinović (see ), while the first inequality of (2.6) by Lazarević (see ).
Now, we will prove that inequalities (2.2) of Theorem 2.1 and (2.4) of Theorem 2.2 may be compared in the following way.
Theorem 2.4. One has
Proof. We must prove the second inequality of (2.7). For this purpose, we will use the inequality (see ):
This implies , so .
Now, we will prove that
By letting , inequality (2.9) becomes
Put , where . After elementary transformations, inequality (2.10) becomes , which is true.
Note. The Referee suggested the following alternative proof: since and the harmonic mean increases in both variables, it suffices to prove stronger inequality which can be written as (2.9).
Remark 2.5. The following refinement of inequalities is true:
Unfortunately, a similar refinement to (2.7) for the mean is not possible, as by numerical examples one can deduce that generally and are not comparable. However, in a particular case, the following result holds true.
Theorem 2.6. Assume that . Then one has
First, prove one the following auxiliary results.
Lemma 2.7. For any , one has
Proof. A computer computation shows that (2.13) is true for . Now put in (2.13). By taking logarithms, the inequality becomes
An easy computation implies
As , we get that for . This means that , as the inequality is true for .
3. Further Improvements
Theorem 3.1. One has
Proof. The inequalities and are proved in . We will see, that further refinements of these inequalities are true. Now, the second inequality of (3.1) follows by the first inequality of (2.3), while the second inequality of (3.2) follows by the first inequality of (2.4). The last inequality is in fact an inequality by Carlson . For the inequalities on we use (2.3) and (2.8).
Theorem 3.3. One has
Proof. The first two inequalities of (3.5) one followed by the first inequality of (3.1) and the fact that with , .
Now, the inequality may be written also as which has been proved in  (see also ).
Further, by Alzer's inequality (see ) one has and by Carlson’s inequality (see ), we get so (3.5) is proved.
The first two inequalities of (3.6) have been proved by the author in . Since (see ) and by (3.2), inequalities (3.6) are completely proved.
Remark 3.4. One has the following inequalities: where ;
Theorem 3.5. One has
Proof. In (3.12), we have to prove the first three inequalities, the rest are contained in (3.5).
The first inequality of (3.12) is proved in . For the second inequality, put By taking logarithms, we have to prove that
As , is strictly increasing, so
The third inequality of (3.12) follows by Carlson's relation (see ).
The first inequality of (3.13) is proved in , while the second one in . The third inequality follows by (see ), while the fourth one by relation (2.9). The fifth one is followed by (2.3).
4. New Huygens Type Inequalities
The main result of this section is contained in the following:
Theorem 4.1. One has
Proof. The first inequalities of (4.1), respectively, (4.2) are the first ones in relations (3.12), respectively, (3.13).
Now, apply the geometric mean-harmonic mean inequality: for , in order to deduce the second inequality of (4.1). The last two inequalities become, after certain transformation,
The proof of (4.2) follows on the same lines, and we omit the details.
Theorem 4.2. For all , one has For all , one has
Remarks 4.3. By (4.1), inequality (4.5) is a refinement of the classical Huygens inequality (1.1):
Similarly, (4.6) is a refinement of the hyperbolic Huygens inequality (1.2):
We will call (4.5) as the second Huygens inequality, while (4.6) as the second hyperbolic Huygens inequality.
In fact, by (4.1) and (4.2) refinements of these inequalities may be stated, too.
The inequality gives
or written equivalently:
The author is indebted to Professor Edward Neuman for his support and discussions on this topic. He also thanks the Referee for a careful reading of the paper and a new proof of Theorem 2.4.
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Copyright © 2012 József Sándor. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.