On Huygens’ Inequalities and the Theory of Means

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Volume 2012, Article ID 597490,9pages doi:10.1155/2012/597490

Research Article

On Huygens’ Inequalities and the Theory of Means

J ´ozsef S ´andor

Department of Mathematics, Babes¸-Bolyai University, Strada Kog˘alniceanu no. 1, 400084 Cluj-Napoca, Romania

Correspondence should be addressed to J´ozsef S´andor,[email protected] Received 24 March 2012; Accepted 20 August 2012

Academic Editor: Mowaﬀaq Hajja

Copyrightq2012 J´ozsef S´andor. 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.

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.

1. Introduction

The famous Huygens’ trigonometric inequalitysee e.g.,1–3states that for allx∈0, π/2 one has

2 sinxtanx >3x. 1.1

The hyperbolic version of inequality1.1has been established recently by Neuman and S´andor3:

2 sinhxtanhx >3x, forx >0. 1.2 Leta, b >0 be two positive real numbers. The logarithmic and identric means ofaand bare defined by

LLa, b: b−a

lnb−lna fora /b; La, a a, IIa, b: 1

e b^b

a^a

_1/b−a

fora /b; Ia, a a,

1.3

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respectively. Seiﬀert’s meanPis defined by

P Pa, b: a−b

2 arcsina−b/ab fora /b, Pa, a a. 1.4 Let

AAa, b: ab

2 , GGa, b

ab,

HHa, b 21

a 1 b

1.5

denote the arithmetic, geometric, and harmonic means ofaandb, 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 assumea /b.

Now, by remarking that lettinga 1sinx,b 1−sinx, wherex ∈ 0, π/2, in P, G,and A, we find that

P sinx

x , Gcosx, A1, 1.6

so Huygens’ inequality1.1may be written also as P > 3AG

2GA32 A 1

G

HA, A, G. 1.7

HereHa, b, cdenotes the harmonic mean of the numbersa, b, c:

Ha, b, c 31 a1

b1 c

. 1.8

On the other hand, by lettingae^x,be^−xinL, G,and A, we find that L sinhx

x , G1, Acoshx, 1.9

so Huygens’ hyperbolic inequality1.2may be written also as L > 3AG

2AG 32 G 1

A

HG, G, A. 1.10

2. First Improvements

Supposea, b >0,a /b.

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Theorem 2.1. One has

P > HL, A> 3AG

2GA HA, A, G, 2.1

L > HP, G> 3AG

2AG HG, G, A. 2.2

Proof. The inequalitiesP > HL, Aand L > HP, Ghave been proved in paper7 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 of2.1, that is, 2LA/LA>

3AG/2GAbecomes, after elementary transformations, exactly inequality1.10, while the second inequality of2.2, that is, 2PG/PG>3AG/2AGbecomes inequality1.7.

Another improvements of1.7, respectively,1.10are provided by Theorem 2.2. One has the inequalities:

P >³

A²G > 3AG

2GA, 2.3

L >³

G²A > 3AG

2AG. 2.4

Proof. The first inequality of2.3is proved in6, while the first inequality of2.8is a well- known inequality due to Leach and Sholander8 see4for many related references. The second inequalities of2.3and2.4are immediate consequences of the arithmetic-geometric inequality applied forA,A,GandA,G,G, respectively.

Remark 2.3. By2.3and1.6, we can deduce the following improvement of the Huygens’

inequality1.1:

sinx x >√³

cosx > 3 cosx

2 cosx1, x∈ 0,π

2 . 2.5

From2.1and1.6, we get sinx

x > 2L^∗

L^∗1 > 3 cosx

2 cosx1, x∈ 0,π

2 .

2.5

Similarly, by2.4and1.9, we get sinhx

x >³

coshx > 3 coshx

2 coshx1, x >0. 2.6

From2.2and1.9, we get sinhx

x > 2P^∗

P^∗1 > 3 coshx

2 coshx1, x >0.

2.6

Here,L^∗L1sinx,1−sinx,P^∗Pe^x, e^−x.

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We note that the first inequality of 2.5 has been discovered by Adamović and Mitrinovićsee3, while the first inequality of2.6by Lazarevićsee3.

Now, we will prove that inequalities 2.2of Theorem 2.1and 2.4of Theorem 2.2 may be compared in the following way.

L >³

G²A > HP, G> 3AG

2AG. 2.7

Proof. We must prove the second inequality of2.7. For this purpose, we will use the inequal- itysee6:

P < 2AG

3 . 2.8

This impliesG/P >3G/G2A, so1/21G/P>2GA/G2A.

Now, we will prove that

2GA G2A > ³

G

A. 2.9

By lettingxG/A∈0,1, inequality2.9becomes 2x1

x2 >√³

x. 2.10

Put x a³, where a ∈ 0,1. After elementary transformations, inequality 2.10 becomesa1a−1³<0, which is true.

Note. The Referee suggested the following alternative proof: sinceP < 2AG/3 and the harmonic mean increases in both variables, it suﬃces to prove stronger inequality√³

A²G >

H2AG/3, Gwhich can be written as2.9.

Remark 2.5. The following refinement of inequalities 2.6

is true:

sinhx x >³

coshx > 2P^∗

P^∗1 > 3 coshx

2 coshx1, x >0. 2.11 Unfortunately, a similar refinement to2.7for the meanPis not possible, as by numeri- cal examples one can deduce that generallyHL, Aand√³

A²Gare not comparable. However, in a particular case, the following result holds true.

Theorem 2.6. Assume thatA/G≥4. Then one has P > HL, A>³

A²G > 3AG

2GA. 2.12

First, prove one the following auxiliary results.

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Lemma 2.7. For anyx≥4, one has

3

x1² 2√³

x−1

> x√³

4. 2.13

Proof. A computer computation shows that2.13is true forx4. Now putxa³in2.13.

By taking logarithms, the inequality becomes fa 2 ln

a³1 2

−9 lna3 ln2a−1>0. 2.14

An easy computation implies a2a−1

a³1 fa 3a−1

a²a−3 . 2.15

As√³ 4²√³

4−3 2√³ 2 √³

2²−3 √³

2−1√³

23 >0, we get thatfa> 0 for a≥√³

4. This means thatfa> f√³

4>0, as the inequality is true fora√³ 4.

Proof of the theorem. We will apply the inequality:

L > ³

G

AG 2

₂

, 2.16

due to the author9. This implies 1 2

1 A

L

< 1 2

1 ³

4A³

GAG²

N. 2.17

By lettingxA/Gin2.13, we can deduce N < ³

A

G. 2.18

So

1 2

1A

L

< ³ A

G. 2.19

This immediately givesHL, A>√³ A²G.

Remark 2.8. If cosx≤1/4,x∈0, π/2, then sinx

x > 2L^∗ L^∗1 >√³

cosx > 3 cosx

2 cosx1, 2.20

which is a refinement, in this case, of inequality 2.5

.

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3. Further Improvements

P >√

LA >³

A²G >AG

L > 3AG

2GA, 3.1

L >√

GP >³

G²A > AG

P > 3AG

2AG. 3.2

Proof. The inequalitiesP > √

LAandL > √

GP are proved in10. We will see, that further refinements of these inequalities are true. Now, the second inequality of3.1follows by the first inequality of2.3, while the second inequality of3.2follows by the first inequality of 2.4. The last inequality is in fact an inequality by Carlson11. For the inequalities onAG/P, we use2.3and2.8.

Remark 3.2. One has sinx

x >√ L^∗>√³

cosx > cosx

L^∗ > 3 cosx

2 cosx1, x∈ 0,π

2 , 3.3

sinhx x >√

P^∗>³

coshx > coshx

P^∗ > 3 coshx

2 coshx1, x >0, 3.4 whereL^∗andP^∗are the same as in

2.6 and

2.5 . Theorem 3.3. One has

P >√

LA > HA, L> AL I > AG

L > 3AG

2GA, 3.5

L > L· I−G A−L >√

IG >√

PG >³

G²A > 3AG

2AG. 3.6

Proof. The first two inequalities of3.5one followed by the first inequality of3.1and the fact thatGx, y> Hx, ywithxL,yA.

Now, the inequalityHA, L> AL/Imay be written also as I > AL

2 , 3.7

which has been proved in4 see also12.

Further, by Alzer’s inequalityL²> GIsee13one has L

I > G

L 3.8

and by Carlson’s inequalityL <2GA/3see11, we get AL

I > AG

L > 3AG

2GA, 3.9

so3.5is proved.

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The first two inequalities of3.6have been proved by the author in5. SinceI > P see14and by3.2, inequalities3.6are completely proved.

Remark 3.4. One has the following inequalities:

sinx x >√

L^∗> 2L^∗ L^∗1 > L^∗

I^∗ > cosx

L^∗ > 3 cosx

2 cosx1, x∈ 0,π

2 , 3.10

whereI^∗I1sinx,1−sinx;

sinhx

x > sinhx x

e^x^coth^x−1−1 coshx−sinhx/x

> e^x^cothx−1/2 >√ P^∗>³

coshx > 3 coshx 2 coshx1.

3.11 Theorem 3.5. One has

P > ³

A

AG 2

₂

>

A

A2G 3

>√

AL > HA, L> AL

I > 3AG

2GA, 3.12

L > ³

G

AG 2

2

>√ IG >

G

2AG 3

>√

PG >³

G²A > 3AG

2AG. 3.13

Proof. In3.12, we have to prove the first three inequalities, the rest are contained in3.5.

The first inequality of3.12is proved in6. For the second inequality, putA/Gt >1 By taking logarithms, we have to prove that

gt 4 ln t1

2

−3 ln t2

3

−lnt >0. 3.14

Asgttt1t2 2t−1>0,gtis strictly increasing, so

gt> g1 0. 3.15

The third inequality of3.12follows by Carlson’s relationL <2GA/3see11.

The first inequality of3.13is proved in9, while the second one in15. The third inequality follows byI > 2AG/3see12, while the fourth one by relation2.9. The fifth one is followed by2.3.

Remark 3.6. The first three inequalities of 3.12 oﬀer a strong improvement of the first inequality of3.1; the same is true for3.13and3.2.

4. New Huygens Type Inequalities

The main result of this section is contained in the following:

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Theorem 4.1. One has P > ³

AAG 2

2

> 3AAG

5AG > A2GA

2AG > 3AG

2GA, 4.1

L > ³

G

AG 2

₂

> 3GAG

5GA > G2AG

2GA > 3AG

2AG. 4.2

Proof. The first inequalities of 4.1, respectively,4.2 are the first ones in relations3.12, respectively,3.13.

Now, apply the geometric mean-harmonic mean inequality:

3

xy²³

x·y·y > 3 1

x 1 y 1

y

3 1

x 2 y

,

4.3

forx A,y AG/2 in order to deduce the second inequality of 4.1. The last two inequalities become, after certain transformation,

A−G² >0. 4.4

The proof of4.2follows on the same lines, and we omit the details.

Theorem 4.2. For allx∈ 0,π

2 , one has

sinx4 tanx

2 >3x. 4.5

For allx >0, one has

sinhx4 tanhx

2 >3x. 4.6

Proof. Apply1.6forP >3AAG/5AGof4.1.

As cosx1 2cos²x/2and sinx 2 sinx/2cosx/2, we get inequality4.5. A similar argument applied to4.6, by an application of4.2and the formulae coshx1 2cosh²x/2and sinhx2 sinhx/2coshx/2.

Remarks 4.3. By4.1, inequality4.5is a refinement of the classical Huygens inequality1.1:

2 sinxtanx >sinx4 tanx

2 >3x.

4.3

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Similarly,4.6is a refinement of the hyperbolic Huygens inequality1.2:

2 sinhxtanhx >sinhx4 tanhx

2 >3x.

4.4

We will call4.5as the second Huygens inequality, while4.6as the second hyperbolic Huygens inequality.

In fact, by4.1and4.2refinements of these inequalities may be stated, too.

The inequalityP > A2GA/2AGgives sinx

x > 2 cosx1

cosx2 , 4.7

or written equivalently:

sinx

x 3

cosx2 >2, x∈ 0,π

2 . 4.8

Acknowledgments

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.

References

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2 J. S. S´andor and M. Bencze, “On Huygens’ trigonometric inequality,” RGMIA Research Report Collection, vol. 8, no. 3, article 14, 2005.

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Applications, vol. 13, no. 4, pp. 715–723, 2010.

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199, no. 2, pp. 629–635, 1996.

10 E. Neuman and J. S´andor, “On the Schwab-Borchardt mean. II,” Mathematica Pannonica, vol. 17, no. 1, pp. 49–59, 2006.

11 B. C. Carlson, “The logarithmic mean,” The American Mathematical Monthly, vol. 79, pp. 615–618, 1972.

12 J. S´andor, “A note on some inequalities for means,” Archiv der Mathematik, vol. 56, no. 5, pp. 471–473, 1991.

13 H. Alzer, “Two inequalities for means,” La Soci´et´e Royale du Canada, vol. 9, no. 1, pp. 11–16, 1987.

14 H. J. Seiﬀert, “Ungleichungen f ¨ur einen bestimmten Mittelwert, Nieuw Arch,” Wisk, vol. 13, no. 42, pp. 195–198, 1995.

15 J. S´andor, “New refinements of two inequalities for means,” submitted.

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