CUSA-HUYGENS, WILKER AND HUYGENS TYPE INEQUALITIES FOR GENERALIZED HYPERBOLIC FUNCTIONS

HAL Id: hal-02527401

https://hal.archives-ouvertes.fr/hal-02527401

Preprint submitted on 1 Apr 2020

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

CUSA-HUYGENS, WILKER AND HUYGENS TYPE INEQUALITIES FOR GENERALIZED HYPERBOLIC

FUNCTIONS

Kwara Nantomah

To cite this version:

Kwara Nantomah. CUSA-HUYGENS, WILKER AND HUYGENS TYPE INEQUALITIES FOR GENERALIZED HYPERBOLIC FUNCTIONS. 2020. �hal-02527401�

INEQUALITIES FOR GENERALIZED HYPERBOLIC FUNCTIONS

KWARA NANTOMAH

This paper is dedicated to those who lost their lives in fighting COVID-19

Abstract. In this paper, we establish Cusa-Huygens, Wilker and Huygens type inequalities for certain generalizations of the hyperbolic functions. From the established results, we recover some previous results as particular cases.

1. Introduction The inequality

sinz

z < cosz+ 2

3 , 0< z < π

2, (1)

is known in the literature as Cusa-Huygens inequality. Its hyperbolic counterpart, which is given as

sinhz

z < coshz+ 2

3 , x >0, (2)

was established by Neuman and Sandor [14]. The inequality sinz

z 2

+tanz

z >2, 0< z < π

2, (3)

which is known as Wilker inequality was first proposed in the classic work [21, p.55] and subsequently attrated the attention of other researchrs. In [23], Wu and Srivastava proved the Wilker-type inequality

z sinz

2

+ z

tanz >2, 0< z < π

2. (4)

The hyperbolic counterpart of (3) was established by Zu [24] as sinhz

z 2

+ tanhz

z >2, z ∈R\ {0}. (5) Also, the hyperbolic counterpart of (4) was established by Wu and Debnath [22]

as

z sinhz

2

+ z

tanhz >2, z ∈R\ {0}. (6)

2010Mathematics Subject Classification. 33B10, 33Bxx, 26D05.

Key words and phrases. Generalized hyperbolic functions, Cusa-Huygens inequality, Wilker inequality, Huygens inequality.

1

2 K. NANTOMAH

Another inequality of interest is the Huygens inequality which is given as [17]

2sinz

z +tanz

z >3, 0< z < π

2, (7)

and its hyperbolic counterpart given as [14]

2sinhz

z +tanhz

z >3, z ∈R\ {0}. (8) Due to their usefulness, these elegant inequalities have been studied extensively and in diverse ways by several researchers. See for example [2], [3], [4], [5], [6], [7], [8], [12], [13], [15], [16], [18], [19], [20], [22], [24], [25], [26] and the related references therein.

Also, in a recent work, the Huygens-type inequality 2

coshz + coshz > sinhz

z + 2tanhz

z > 1

coshz + 2, z ∈R\ {0}, (9) was established among other things by Bagul and Chesneau [1].

Motivated by the results (2), (5), (6), (8) and (9), the objective of this paper is to establish analogous inequalities concerning certain generalizations of the hy- perbolic functions. The established results serve as generalizations of the previous results.

2. Preliminary Definitions

In a bid to generalized a previous work [9], the authors of [10] gave the following generalizations of the hyperbolic functions.

Definition 2.1. The generalized hyperbolic cosine, hyperbolic sine and hyper- bolic tangent functions are respectively defined as [10]

cosh_a(z) = a^z+a^−z

2 , (10)

sinh_a(z) = a^z−a^−z

2 , (11)

tanh_a(z) = sinh_a(z)

cosh_a(z) = a^z−a^−z

a^z +a^−z = 1− 2

1 +a^2z, (12) where a >1 and z ∈R.

These generalized functions satisfy the following identities.

cosh_a(z) + sinh_a(z) =a^z, (13) cosh_a(z)−sinh_a(z) =a^−z, (14) (cosh_a(z))⁰ = (lna) sinh_a(z), (15) (sinh_a(z))⁰ = (lna) cosh_a(z), (16)

(tanha(z))⁰ = lna

cosh²_a(z), (17)

(cosh_a(z))⁰⁰+ (sinh_a(z))⁰⁰ = (lna)²a^z, (18) (cosh_a(z))⁰⁰−(sinh_a(z))⁰⁰ = (lna)²a^−z, (19)

cosh²_a(z) + sinh²_a(z) = cosha(2z), (20) cosh²_a(z)−sinh²_a(z) = 1, (21) 2 sinh_a(z) cosh_a(z) = sinh_a(2z), (22)

cosh²_a(z) = cosh_a(2z) + 1

2 , (23)

sinh²_a(z) = cosha(2z)−1

2 . (24)

The generalized hyperbolic secant, hyperbolic cosecant and hyperbolic cotangent functions are respectively defined as

sech_a(z) = 1

cosh_a(z), cosech_a(z) = 1

sinh_a(z), coth_a(z) = 1

tanh_a(z). (25) As pointed out in [10], several other identities can be derived from (10), (11) and (12). When a = e, where e = 2.71828... is the Euler’s number, then the above definitions and indentities reduce to their ordinary counterparts.

3. Results and Discussion Lemma 3.1. The inequality

cosh_a(z)<

sinh_a(z) z

3

, (26)

holds for z ∈R\ {0}.

Proof. Inequality (26) has been proved in [11] for z > 0. Now let z < 0 so that

−z >0. Then

cosh_a(z) = cosh_a(−z)<

sinh_a(−z)

−z 3

=

sinh_a(z) z

3

,

which completes the proof.

Since the function ^sinh_z^a(z) is increasing for z >0 and decreasing forz <0, then Lemma3.1 implies the following generalized result.

Lemma 3.2. The inequality

cosha(z)<

sinh_a(z) z

v

, (27)

holds for z ∈R\ {0} and v ≥3.

Lemma 3.3 ([11]). For z ∈R\ {0}, the inequality lna

cosh_a(z) < sinh_a(z)

z <(lna) cosh_a(z), (28)

holds.

Lemma 3.4 (Young’s Inequality). Let x, y ≥0, a, b∈(0,1) such that a+b = 1.

Then,

x^ay^b ≤ax+by. (29)

4 K. NANTOMAH

Theorem 3.5. The inequality sinh_a(z)

z < 2 lna+ (lna) cosh_a(z)

3 , (30)

holds for z ∈R\ {0}.

Proof. Since the functions in each term of the inequality are even, it suffices to prove the case forz >0. Let z >0 and h be defined as

h(z) = 2z+zcosh_a(z) sinh_a(z) . Then

h⁰(z) = 1

sinh²_a(z)[2 sinh_a(z) + cosh_a(z) sinh_a(z)−2z(lna) cosh_a(z)−(lna)z]

= 1

sinh²_a(z)φ(z) and then

φ⁰(z) = (lna) [cosh_a(2z)−2(lna)zsinh_a(z)−1]

= 2(lna) sinh_a(z) [sinh_a(z)−(lna)z]>0,

since ^sinh_z^a(z) > lna for z ∈ (0,∞). Hence φ(z) is increasing and consequently, φ(z)> φ(0) = 0. Thus h(z) is increasing. Hence

h(z)>lim

z→0h(z) = 3 lna,

which gives (30).

Remark 3.6. When a=e, then inequality (30) reduces to the hyperbolic Cusa- Huygens inequality (2).

Theorem 3.7. The inequalities sinh_a(z)

z 2

+tanh_a(z)

z >2, (31)

z sinh_a(z)

2

+ z

tanh_a(z) > 1 + lna

(lna)² , 1< a≤e, (32) hold for z ∈R\ {0}.

Proof. Let z ∈ R\ {0}. Then by the AM-GM inequality and Lemma 3.1, we obtain

sinh_a(z) z

2

+tanh_a(z)

z ≥2

s

sinh_a(z) z

2

tanh_a(z) z

= 2 s

sinh_a(z) z

3

1 cosh_a(z)

>2,

which gives (31). To prove (32), it suffices to prove the case forz >0. Let z >0 and define ψ(z) by

ψ(z) =

z sinh_a(z)

2

+ z

tanh_a(z),

where 1 < a≤ e. Then by differentiating, applying the AM-GM inequality and Lemma3.1, we obtain

ψ⁰(z)

= 1

sinh³_a(z)

sinh²_a(z) cosh_a(z) + (2−lna)zsinh_a(z)−2(lna)z²cosh_a(z)

≥ 1

sinh³_a(z)

2 q

sinh²_a(z) cosh_a(z).(2−lna)zsinh_a(z)−2(lna)z²cosh_a(z)

= 2z² sinh³_a(z).p

cosh_a(z)





√2−lna s

sinh_a(z) z

3

−(lna)p

cosh_a(z)





>0.

Thusψ(z) is increasing. Hence ψ(z)> lim

z→0ψ(z) = 1 + lna (lna)² ,

which gives (32).

Remark 3.8. When a=e, then inequalities (31) and (31) reduce to the hyper- bolic Wilker-type inequalities (5) and (6) respectively.

Theorem 3.9. The inequality sinh_a(z)

z 2

+tanh_a(z) z >

z sinh_a(z)

2

+ z

tanh_a(z), (33) holds for z ∈R\ {0}.

Proof. Observe that ^A1^2+B

A2+_B¹ =A²B. Then by Lemma 3.1, we obtain sinha(z)

z

2

+ ^tanh_z^a(z) z

sinha(z)

2

+ _tanh^z

a(z)

=

sinha(z) z

2

tanha(z) z

=

sinha(z) z

3

1 cosh_a(z)

>1,

which concludes the proof.

Theorem 3.10. Let α, β ∈(0,1) such that α+β= 1. Then the inequality α

sinh_a(z) z

2

+β

tanh_a(z) z

>(lna)^2(α−β), (34)

holds for z ∈R\ {0}

6 K. NANTOMAH

Proof. Let z ∈R\ {0}. Then Youngs inequality (29) and Lemma 3.1 imply that α

sinha(z) z

2

+β

tanha(z) z

≥

sinha(z) z

2α

tanha(z) z

β

=

sinh_a(z) z

2α+β 1 cosh_a(z)

β

>

sinh_a(z) z

2α+β

sinh_a(z) z

−3β

=

sinh_a(z) z

2(α−β)

>(lna)^2(α−β),

which completes the proof.

Remark 3.11. If α=β = ¹₂, then (34) reduces to (31).

Theorem 3.12. The inequality

2sinh_a(z)

z + tanh_a(z)

z >3 lna, (35)

holds for z ∈R\ {0}.

Proof. It suffices to prove the case for z >0. Let z >0 and let h be defined as h(z) = 2sinh_a(z)

z + tanh_a(z)

z .

Then

z²h⁰(z) = 2(lna)zcosh_a(z)−2 sinh_a(z) + (lna)zsech²_a(z)−tanh_a(z)

=θ(z), and

θ⁰(z) = 2(lna)²zsinh_a(z)−2(lna)²ztanh_a(z)sech²_a(z)

= 2(lna)²zsinh_a(z)

1− 1

cosh³_a(z)

>0,

which shows that θ(z) is increasing. Hence θ(z) > θ(0) = 0. Thus, h(z) is increasing and consequently,

h(z)>lim

z→0h(z) = 3 lna

which yields (35).

Remark 3.13. When a =e, then inequality (35) reduces to (8).

Theorem 3.14. The inequality

2 z

sinh_a(z)+ z

tanh_a(z) > 3

lna, (36)

holds for z ∈R\ {0}.

Proof. It suffices to prove the case for z >0. Let z >0 and δ be defined as δ(z) = 2 z

sinh_a(z) + z tanh_a(z). Then

δ⁰(z) = 1

sinh²_a(z)[2 sinh_a(z)−zlna+ cosh_a(z)(sinh_a(z)−zlna)]>0 since sinh_a(z)> zlna. Hence δ(z) is increasing and consequently,

δ(z)>lim

z→0δ(z) = 3 lna

which yields (36).

Theorem 3.15. The inequality

(m+ 1)sinh_a(z)

z +mtanh_a(z)

z > msinh_a(z)

z + (m+ 1)tanh_a(z)

z , (37)

holds for z ∈R\ {0} and m ∈N∪ {0}.

Proof. z ∈R\ {0} . Then sinh_a(z)

z − tanh_a(z)

z = sinh_a(z) z

1− 1

cosh_a(z)

>0, since cosh_a(z)>1 forz ∈R. That is

sinh_a(z)

z > tanh_a(z)

z . (38)

Adding m

sinha(z)

z + ^tanh_z^a(z)

to both sides of (38) completes the proof.

Theorem 3.16. The inequality 2 lna

cosha(z) + (lna) cosh_a(z)> sinh_a(z)

z + 2tanh_a(z)

z > lna

cosha(z) + 2 lna, (39) holds for z ∈R\ {0}.

Proof. It suffices to prove the case for z >0. Let z >0 and f be defined as f(z) = 2(lna)z+ (lna)zcosh²_a(z)−sinh_a(z) cosh_a(z)−2 sinh_a(z).

Then

f⁰(z) = 2(lna) + 2(lna)²zcosh_a(z) sinh_a(z)−(lna) sinh²_a(z)−2(lna) cosh_a(z), and then

f⁰⁰(z) = 2(lna)³zsinh²_a(z) + 2(lna)²

(lna)zcosh²_a(z)−sinh_a(z)

>0, since sinh_a(z)<(lna)zcosh_a(z) (see Lemma3.3) and cosh_a(z)<cosh²_a(z). Hence f⁰(z) is increasing and so, f⁰(z) > f⁰(0) = 0. Thus, f(z) is increasing and so f(z)> f(0) = 0. This yields the left-hand side of (39). Next, forz >0, let g be defined as

g(z) = sinh_a(z) cosh_a(z) + 2 sinh_a(z)−(lna)z−2(lna)zcosh_a(z).

8 K. NANTOMAH

Then

g⁰(z) = (lna) cosh²_a(z) + (lna) sinh²_a(z)−lna−2(lna)²zsinh_a(z) and then

g⁰⁰(z) = 2(lna)²cosh_a(z) [sinh_a(z)−(lna)z] + 2(lna)²sinh_a(z) [cosh_a(z)−1]

>0,

since sinh_a(z) > (lna)z and cosh_a(z) > 1 . Hence g⁰(z) is increasing and so, g⁰(z) > g⁰(0) = 0. Thus, g(z) is increasing and so g(z) > g(0) = 0. This yields the right-hand side of (39) and that completes the proof.

Remark 3.17. When a =e, then inequality (39) reduces to (9).

Remark 3.18. By (37) and (39), we have 2sinh_a(z)

z +tanh_a(z)

z > lna

cosha(z) + 2 lna.

This is however weaker than inequality (35).

References

[1] Y. J. Bagul and C. Chesneau Two double sided inequalities involving sinc and hyperbolic sinc functions, Int. J. Open Problems Compt. Math., 12(4)(2019), 15-20.

[2] Y. J. Bagul and C. Chesneau Some New Simple Inequalities Involving Exponential, Trigonometric and Hyperbolic Functions, CUBO A Mathematical Journal, 21(1)(2019), 21-35.

[3] C-P. ChenWilker and Huygens Type Inequalities for the Lemniscate Functions, J. Math.

Inequal., 6(4)(2012), 673-684.

[4] C-P. Chen and W-S. CheungSharp Cusa and Becker-Stark inequalities, J. Inequal. Appl., 2011(2011): 136.

[5] C-P. Chen and J. Sandor Inequality chains for Wilker, Huygens and Lazarevic type in- equalities, J. Math. Inequal., 8(1)(2018), 55-67.

[6] L-G. Huang, L. Yin, Y-L. Wang and X-L. LinSome Wilker and Cusa type inequalities for generalized trigonometric and hyperbolic functions, J. Inequal. Appl., 2018(2018), 1-8, Art ID: 52.

[7] B. Malesevic, T. Lutovac, M. Rasajski and C. MorticiExtensions of the natural approach to refinements and generalizations of some trigonometric inequalities, Advances in Difference Equations 2018(2018) Article ID: 90, 15 pages.

[8] C. Mortitci The natural approach of Wilker-Cusa-Huygens inequalities, Math. Inequal.

Appl., 14(2011), 535-541.

[9] K. NantomahOn Some Properties of the Sigmoid Function, Asia Mathematika, 3(1)(2019), 79-90.

[10] K. Nantomah, C. A. Okpoti and S. NasiruOn a Generalized Sigmoid Function and its Prop- erties, Asian Journal of Mathematics and Applications, 2020(2020), Article ID ama0527, 11 pages

[11] K. Nantomah and E. Prempeh Some Inequalities for Generalized Hyperbolic Functions.

2020. hal-02524105f.https://hal.archives-ouvertes.fr/hal-02524105

[12] E. NeumanOn Wilker and Huygens type inequalities, Math. Inequal. Appl., 15(2) (2012), 271-279.

[13] E. Neuman Wilker and Huygens- type inequalities for the generalized trigonometric and for the generalized hyperbolic functions, Appl. Math. Comput., 230(2014), 211-217.

[14] E. Neuman and J. Sandor On some inequalities involving trigonometric and hyperbolic functions with emphasis on the Cusa-Huygens, Wilker, and Huygens inequalities, Math.

Inequal. Appl., 13(4)(2010), 715-723.

[15] J. Sandor On Huygens Inequalities and the Theory of Means, Int. J. Math. Math. Sci., 2012(2012), Article ID 597490, 9 pages.

[16] J. SandorSharp CusaHuygens and related inequalities, Notes on Number Theory and Dis- crete Mathematics, 19(1)(2013), 50-54.

[17] J. Sandor and M. BenczeOn Huygens trigonometric inequality, RGMIA Res. Rep. Coll.

8(3)(2005), Art. 14.

[18] J. Sandor and R. Olah-GalOn Cusa-Huygens type trigonometric and hyperbolic inequali- ties, Acta Univ. Sapientiae, Mathematica, 4(2)(2012), 145-153.

[19] J. S. Sumner, A. A. Jagers, M. Vowe and J. AnglesioInequalities involving trigonometric function, Am. Math. Mon., 98(3)(1991), 264-267.

[20] Z. Sun and L. ZhuOn New Wilker-Type Inequalities, ISRN Math. Anal., 2011(2011), 1-7.

[21] J. B. WilkerElementary Problems: E3301-E3306, Am. Math. Mon., 96(1)(1989), 54-55.

[22] S. Wu and L. DebnathWilker-type inequalities for hyperbolic functions, Appl. Math. Lett., 25(2012), 837-842.

[23] S-H. Wu and H. M. Srivastava A weighted and exponential generalization of Wilkers in- equality and its applications, Integral Transforms Spec. Funct., 18(7-8)(2007), 529-535.

[24] L. ZhuOn Wilker-type Inequalities, Math. Inequal. Appl., 10(4)(2007), 727-731.

[25] L. Zhu A source of inequalities for circular functions, Comput. Math. Appl., 58(2009), 1998-2004.

[26] L. Zhu New inequalities of Wilker’s type for hyperbolic functions, AIMS Mathematics, 5(1)(2019), 376-384.

Department of Mathematics, Faculty of Mathematical Sciences, University for Development Studies, Navrongo Campus, P. O. Box 24, Navrongo, UE/R, Ghana.

E-mail address: [email protected]