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.


Introduction
The famous Huygens' trigonometric inequality see e.g., 1-3 states that for all x ∈ 0, π/2 one has 2 sin x tan x > 3x. 1.1 The hyperbolic version of inequality 1.1 has been established recently by Neuman and Sándor 3 : denote the arithmetic, geometric, and harmonic means of a and b, 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 a / b.Now, by remarking that letting a 1 sin x, b 1 − sin x, where x ∈ 0, π/2 , in P, G, and A, we find that so Huygens' inequality 1.1 may be written also as Here H a, b, c denotes the harmonic mean of the numbers a, b, c: On the other hand, by letting a e x , b e −x in L, G, and A, we find that so Huygens' hyperbolic inequality 1.2 may be written also as

First Improvements
Suppose a, b > 0, a / b.
Proof.The first inequality of 2.3 is proved in 6 , while the first inequality of 2.8 is a wellknown inequality due to Leach and Sholander 8 see 4 for many related references .The second inequalities of 2.3 and 2.4 are immediate consequences of the arithmetic-geometric inequality applied for A, A, G and A, G, G, 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, L * L 1 sin x, 1 − sin x , P * P e x , e −x .

International Journal of Mathematics and Mathematical Sciences
We note that the first inequality of 2.5 has been discovered by Adamović and Mitrinović see 3 , while the first inequality of 2.6 by Lazarević see 3 .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 6 : This implies G/P > 3G/ G 2A , so 1/2 1 G/P > 2G A / G 2A .Now, we will prove that By letting x G/A ∈ 0, 1 , inequality 2.9 becomes 2x 1
Note.The Referee suggested the following alternative proof: since P < 2A G /3 and the harmonic mean increases in both variables, it suffices to prove stronger inequality 3

√
A 2 G > H 2A G /3, G which can be written as 2.9 .
Remark 2.5.The following refinement of inequalities 2.6 is true: Unfortunately, a similar refinement to 2.7 for the mean P is not possible, as by numerical examples one can deduce that generally H L, A and 3

√
A 2 G are not comparable.However, in a particular case, the following result holds true.Theorem 2.6.Assume that A/G ≥ 4. Then one has First, prove one the following auxiliary results.

2.13
Proof.A computer computation shows that 2.13 is true for x 4. Now put x a 3 in 2. 13 .By taking logarithms, the inequality becomes An easy computation implies Proof of the theorem.We will apply the inequality:

2.16
due to the author 9 .This implies

2.17
By letting x A/G in 2.13 , we can deduce

2.19
This immediately gives H L, A > which is a refinement, in this case, of inequality 2.5 .

Further Improvements
Theorem 3.1.One has Proof.The inequalities P > √ LA and L > √ GP are proved in 10 .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 11 .For the inequalities on AG/P, we use 2.3 and 2.8 .
where L * and P * are the same as in 2.6 and 2.5 .

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 G x, y > H x, y with x L, y A. Now, the inequality H A, L > AL/I may be written also as which has been proved in 4 see also 12 .Further, by Alzer's inequality L 2 > GI see 13 one has and by Carlson's inequality L < 2G A /3 see 11 , we get The first two inequalities of 3.6 have been proved by the author in 5 .Since I > P see 14 and by 3.2 , inequalities 3.6 are completely proved.Remark 3.4.One has the following inequalities: where

3.11
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 6 .For the second inequality, put A/G t > 1 By taking logarithms, we have to prove that g t 4 ln t 1 2 − 3 ln t 2 3 − ln t > 0.

3.14
As g t t t 1 t 2 2 t − 1 > 0, g t is strictly increasing, so g t > g 1 0.

3.15
The third inequality of 3.12 follows by Carlson's relation L < 2G A /3 see 11 .
The first inequality of 3.13 is proved in 9 , while the second one in 15 .The third inequality follows by I > 2A G /3 see 12 , while the fourth one by relation 2.9 .The fifth one is followed by 2.3 .
Remark 3.6.The first three inequalities of 3.12 offer a strong improvement of the first inequality of 3.1 ; the same is true for 3.13 and 3.2 .

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:

2e b b a a 1 / 1 . 3 International
sinh x tanh x > 3x, for x > 0. 1.2 Let a, b > 0 be two positive real numbers.The logarithmic and identric means of a and b are defined by L L a, b : b − a ln b − ln a for a / b ; L a, a a, I I a, b : 1 b−a for a / b ; I a, a a, Journal of Mathematics and Mathematical Sciences respectively.Seiffert's mean P is defined by P P a, b : a − b 2 arcsin a − b / a b for a / b , P a, a a.

Theorem 2.2. One has the inequalities:
Proof.Apply 1.6 for P > 3A A G / 5A G of 4.1 .As cos x 1 2cos 2 x/2 and sin x 2 sin x/2 cos x/2 , we get inequality 4.5 .A similar argument applied to 4.6 , by an application of 4.2 and the formulae cosh x 1 2cosh 2 x/2 and sinh x 2 sinh x/2 cosh x/2 .Remarks 4.3.By 4.1 , inequality 4.5 is a refinement of the classical Huygens inequality 1.1 :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.