Some Inequalities for the Generalized Trigonometric

Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 3, 96–101 Available online at http://pubs.sciepub.com/tjant/2/3/8 c Science and Education Publishing

DOI: 10.12691/tjant-2-3-8

Some Inequalities for the Generalized Trigonometric and Hyperbolic Functions Li Yin1,∗ , Li-Guo Huang2 , and Feng Qi3,4,5 1,2

Department of Mathematics, Binzhou University, Binzhou City, Shandong Province, 256603, China E-mail: yinli [email protected], [email protected] 3 College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China 4 Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China 5 Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China ∗ Corresponding author: yinli [email protected]

Received June 21, 2014; Revised July 01, 2014; Accepted July 13, 2014 Abstract In the paper, the authors establish some inequalities of the generalized trigonometric and hyperbolic functions, partially solve a conjecture posed by R. Klén, M. Vuorinen, and X.-H. Zhang, and finally pose an open problem. Keywords: inequality, generalized trigonometric function, generalized hyperbolic functions, Bernoulli inequality, Huygens’ inequality, Wilker’s inequality, conjecture, open problem MSC: Primary 33B10; Secondary 26A51, 26D05, 33C20 Cite This Article: Li Yin, Li-Guo Huang, and Feng Qi, “Some inequalities for the generalized trigonometric and hyperbolic functions.” Turkish Journal of Analysis and Number Theory, vol. 2, no. 3 (2014): 96–101. doi: 10.12691/tjant-2-3-8.

1. Introduction It is well known from calculus that Z x 1 arcsin x = dt (1 − t2 )1/2 0 and π = arcsin 1 = 2

Z 0

1

1 dt (1 − t2 )1/2

(1.1)

(1.2)

for 0 ≤ x ≤ 1. For 1 < p < ∞ and 0 ≤ x ≤ 1, the arc sine may be generalized as Z x 1 arcsinp x = dt (1.3) (1 − tp )1/p 0

where Γ denotes the classical gamma function. Hence, 2π we have πp = p sin(π/p) . π The inverse of arcsinp x on 0, 2p is called the generalized sine function, denoted by sinp x, and may be extended to (−∞, ∞). See [7] and closely related references therein. π For x ∈ 0, 2p , the generalized cosine function cosp x is defined by d sinp x . (1.5) cosp x = dx It is easy to see that 1/p cosp x = 1 − sinpp x (1.6) and

and Z

1

πp 1 π π = arcsinp 1 = d t = csc p 1/p 2 p p 0 (1 − t ) Z ∞ 1/p−1 1 1 1 1 t = Γ d t, Γ 1− = p p p p 0 1+t

(1.4)

d cosp x = − cos2−p x sinpp−1 x. (1.7) p dx The generalized tangent function tanp x is defined as sinp x πp tanp x = , x ∈ R \ kπp + : k ∈ Z . (1.8) cosp x 2

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Turkish Journal of Analysis and Number Theory

From (1.8), it follows that

and

p d tanp x = 1 + tanp x , dx

x∈

πp πp − , . 2 2

(1.9)

The generalized secant function secp x is defined as 1 πp secp x = , x ∈ 0, . (1.10) cosp x 2 It follows from (1.8) and (1.9) that πp p p secp x = 1 + tanp x, x ∈ 0, 2

(1.11)

d sechp x = − sechp x tanhpp−1 x. (1.21) dx The above formulas (1.5) to (1.9) and (1.15) to (1.19) can be found in [8]. For r > 0, the set S2 = {(x, y) ∈ R2 : x2 − y 2 = r2 } is the equilateral hyperbola in the plane R2 with the `2 metric. The connection between the hyperbolic coordinates (r, φ) and the rectangular coordinates (x, y) is given by x = r cosh φ and y = r sinh φ. We may easily obtain that, when x, y > 0, x2 − y 2 = r2 and φ are related by φ = arctanh xy . When p 6= 2, the analogue of the equilateral hyperbola is the p-equilateral hyperbola

and

Sp = {(x, y) ∈ R2 : xp − y p = rp } d secp x = secp x tanp−1 x, p dx

πp x ∈ 0, . 2

(1.12)

The generalized cosecant function cscp x may be defined as πp 1 , x ∈ 0, . (1.13) cscp x = sinp x 2 It is clear that 1 d cscp x cscp x and =− (1.14) tanpp x dx tanp x π for x ∈ 0, 2p . The generalized inverse hyperbolic sine function arcsinhp x is defined by Z x 1  d t, x ∈ [0, ∞), p 1/p arcsinhp x = 0 (1 + t )  − arcsinhp (−x), x ∈ (−∞, 0). (1.15) The inverse of arcsinhp x is called the generalized hyperbolic sine function and denoted by sinhp x. The generalized hyperbolic cosine function coshp x is defined as d sinhp x coshp x = . (1.16) dx It is easy to show that cscpp x = 1 +

coshpp x − | sinhp x|p = 1,

x∈R

(1.17)

and d coshp x = cosh2−p x sinhp−1 x, p p dx

x ≥ 0.

(1.18)

The generalized hyperbolic tangent function and the generalized hyperbolic secant function are defined as tanhp x =

sinhp x coshp x

and

(1.22)

sechp x =

1 . (1.19) coshp x

Their derivatives are d tanhp x = 1 − tanhpp x = sechpp x dx

(1.20)

and the identities x = r coshp φ and y = r sinhp φ hold. From this, it follows that |coshp φ|p − |sinhp φ|p = 1 and 1/p coshp φ = 1 + (sinhp φ)p ,

x, y > 0.

This gives a geometrical interpretation to sinhp x and coshp x. Furthermore, we may also define the generalized trigonometric functions by means of Gauss hypergeometric function. Interested readers may refer to [1]. As well as we known, the “hyperbolic” function were introduced in 1760 independent by Vincenzo Riccati and John Heinrid Lambert, the notations sh and ch are still used in some other languages such as European, French, and Russian. The hyperbolic function occurs in the solutions of some linear differential equations, such as defining catenary, Laplaces equations in Cartesian coordinates, and occurs in many important areas in physics, such as special relativity. In complex analysis, the hyperbolic function arises the imaginary parts of sine and cosine. For complex variables, the hyperbolic functions are rational functions of exponential functions and are meromorphic. Therefore, many advantages properties relating to the hyperbolic function have already been applied extensively. For more information on this topic, please read the classical book [5]. During the last decades, many authors have studied the generalized trigonometric functions introduced in [9]. See, for example, [1, 2, 3, 4, 7] and plenty of references therein. In [8], some classical inequalities for generalized trigonometric and hyperbolic functions, such as Mitrinović-Adamović inequality, Huygens’ inequality, and Wilker’s inequality were generalized. In [6], some basic properties of the generalized (p, q)-trigonometric functions were given. Recently, the functions arcsinp,q x and arcsinhp,q x were expressed in terms of Gaussian hypergeometric functions and many properties and inequalities for generalized trigonometric and hyperbolic functions were established in [3]. In [1], some Tur´ an


type inequalities for generalized trigonometric and hyperbolic functions were presented. Very recently, a conjecture posed in [3] was verified in [7]. In this paper, we will establish some inequalities of the generalized trigonometric and hyperbolic functions, partially solve a conjecture in [8], and finally pose an open problem.

Lemma 2.7 ([8, Theorems 3.6 and 3.7]). For all p ∈ (1, ∞), we have sinp x πp α cosp x < < 1, x ∈ 0, (2.8) x 2 and coshα p x
1, then p sinp x tanp x + > 1 + p, x x

πp 0 0,

where g(x) = coshp x sinp x − sinhp x cosp x.

(2.1)

Lemma 2.3 ([8, Theorem 3.4]). For p ∈ [2, ∞) and π x ∈ 0, 2p , we have

1 p+1

x > 0,

π Lemma 2.8. For p > 1 and x ∈ 0, 2p , the function sinh x f (x) = sinppx is positive and strictly increasing.

g (x) =

(1 + t)α > 1 + αt.

sinhp x < coshβp x, x

where the constants α = possible.

0

Lemma 2.2 ([10, Bernoulli inequality]). For t > −1 and α > 1, we have

98

This means that g(x) is strictly increasing and g(x) > g(0) = 0. Hence, it follows that f 0 (x) > 0 and that f (x) is strictly increasing. π Lemma 2.9. For p > 1 and x ∈ 0, 2p , the functions g1 (x) = sinp x − x cosp x and g2 (x) = x coshp x − sinhp x are positive. Proof. An easy computation yields g10 (x) = x cosp x tanp−1 x>0 p

(2.3) and

and p sinhp x tanhp x + > 1 + p, x x

g20 (x) = x coshp x tanhp−1 x > 0. p x > 0.

(2.4)

Lemma 2.5 ([8, Theorem 3.22]). For p ∈ (1, 2], the double inequality sinp x cosp x + p cosp x + 2 < ≤ x 1+p 3 π holds for all x ∈ 0, 2p .

(2.5)

Lemma 2.6 ([8, Theorem 3.24]). For all x > 0, 1. if p ∈ (1, 2], then sinhp x coshp x + p < ; x 1+p

3. Main Results Now we are in a position to present our main results. π Theorem 3.1. For p > 1 and x ∈ 0, 2p , we have p sinp x tanp x + > 2. (3.1) x x Proof. Setting t = in (2.1) leads to

(2.6)

2. if p ∈ (2, ∞), then sinhp x coshp x + 2 < . x 3

These imply that g1 (x) > g1 (0) = 0 and that g2 (x) > g2 (0) = 0.

(2.7)

sinp x x

p

sinp x x

− 1 ∈ (−1, 0) and α = p

sinp x >1+p −1 x tanp x tanp x >1−p+1+p− =2− . x x

Further using (2.3) results in the inequality (3.1).

99


Remark 3.1. The inequality (3.1) is an analogy of Wilker’s inequality involving the sine and tangent functions. See [12, Section 8.1]. π Theorem 3.2. For p ∈ [2, ∞) and x ∈ 0, 2p , we have p tanp x x + > 2. (3.2) sinhp x x Proof. Letting t = sinhxp x − 1 and α = p in (2.1) gives p x x ≥1−p+p sinhp x sinhp x sinp x tanp x >1−p+p >2− . x x Combining this with (2.2) and (2.3) yields (3.2). π Theorem 3.3. For p ∈ (1, 2] and x ∈ 0, 2p , we have px x + > 1 + p. (3.3) sinp x tanp x Proof. This follows from using (2.5) and p + cosp x − (1 + p)

Theorem 3.6. For p ∈ (1, ∞) and x ∈ 0, have p p tanp x x sinp x x + + > . x x sinp x tanp x

> p + cosp x − (1 + p)

cosp x + p = 0. 1+p

Theorem 3.4. For p ∈ (1, 2] and x > 0, we have x px + > 1 + p. (3.4) sinhp x tanhp x

coshp x + p = 0. 1+p

Theorem 3.5. For p ∈ (1, 2], we have p x πp x + > 2, x ∈ 0, sinp x tanp x 2 x sinhp x

+

p+1

1 > 1. cosp x

This can be derived from utilizing (2.8) as follows

sinp x x

p+1

p+1 1 1 > cosp1/(p+1) x = 1. cosp x cosp x

Theorem 3.7. For p ∈ (1, ∞) and x > 0, we have p p sinhp x x tanhp x x > . (3.8) + + x x sinhp x tanhp x

Remark 3.3. Inequalities (3.7) and (3.8) imply inequalities (3.1) and (3.20) in [8, Corollary 3.19]. π Theorem 3.8. For p > 1 and x ∈ 0, 2p , we have tanp x x > . x sinp x

(3.9)

f 0 (x) = sinp x secpp x + sinp x − 2x,

x > 2, x > 0. tanhp x

(3.5)

(3.6)

Proof. Taking t = sinxp x − 1 and α = p in (2.1) and using the inequality (3.3) result in p x x >1+p −1 sinp x sinp x x x >1−p+1+p− =2− . tanp x tanp x The inequality (3.6) may be deduced similarly.

f 00 (x) = cosp x secpp x + p sinp x secpp x tanp−1 x p + cosp x − 2, f (x) = (2p − 1) secpp x − 1 sinpp−1 x secp−2 x p 000

+ p2 sinp2p−1 x secp3p−2 x

and p

sinp x x

Proof. Let f (x) = tanp x sinp x − x2 . An easy computation yields

sinhp x x

Remark 3.2. Inequalities presented in Theorems 3.3 and 3.4 are analogies of Huygens’ inequality for the sine and tangent functions. See [11].

(3.7)

tan x sinp x p and b = xp , Proof. When letting a = x 1 1 the inequality (3.7) becomes a + b > a + b which is equivalent to ab > 1, that is,

Proof. This follows from using (2.6) and

> p + coshp x − (1 + p)

we

Proof. This follows from using the inequality (2.9).

sinp x x

p + coshp x − (1 + p)

πp 2 ,

+ p(p − 1) sinpp−1 x secp3p−2 x ≥ 0. Hence, f 00 (x) > f 00 (0) = 0, f 0 (x) is strictly increasing, f 0 (x) > f 0 (0) = 0, and f (x) > f (0) = 0. The inequality (3.9) follows. Theorem 3.9. For p > 1 and x ∈ (0, ∞), we have tanhp x p+1 > x p coshp x + 1

(3.10)

or, equivalently, sinhp x (p + 1) coshp x > . x p coshp x + 1

(3.11)

Turkish Journal of Analysis and Number Theory Proof. Set f (x) = (p + 1)x − tanhp x p coshp x + 1 . Then f 0 (x) = p − p coshp x + tanhpp x

Theorem 3.11. For p > 1 and x ∈ (0, 1), we have 1

Z 0

and

Accordingly, it follows that f 0 (x) < f 0 (0) = 0, f (x) is strictly decreasing, and f (x) < f (0) = 0. The inequalities (3.10) and (3.11) are proved. Theorem

3.9

(3.12)

Theorem 3.10. For p ≥ 3 and x ∈ x∗p , function f (x) =

ln ln

x sinp x sinhp x x

and

πp 2 ,

the

is strictly increasing, where x∗p

satisfies the equation coshpp x = p − 2.

1

Z 0

0

xp−2 sinp x √ d x. p 1 − xp

(3.14)

cosp x √ dx = p 1 − xp

πp /2

Z

cosp sinp t d t.

(3.15)

0

Similarly, taking t = arccosp x, the right hand side of (3.14) reduces to Z

sinhp x p+1 > . x p + coshp x

1

Z

cosp x √ dx > p 1 − xp

Proof. If putting t = arcsinp x, the left hand side of (3.14) becomes

f 00 (x) = p tanhp−1 x sechpp x − coshp x < 0. p

Remark 3.4. Considering coshp x > 1 reveals

100

0

1

xp−2 sinp x √ dx = p 1 − xp

Z

πp /2

sinp cosp t d t.

(3.16)

0

Making use of the monotonicity of sinp x and cosp x acquires cosp sinp t > cosp t > sinp cosp t. The inequality (3.14) is thus proved.

sinh x

Proof. Let f1 (x) = ln sinxp x and f2 (x) = ln xp . Then f1 (0) = limx→0 f1 (x) = 0, f2 (0) = limx→0 f2 (x) = 0, and f10 (x) f20 (x)

=

sinhp x g1 (x) , sinp x g2 (x)

(3.13)

where g1 (x) = sinp x − x cosp x and g2 (x) = x coshp x − sinhp x satisfy g1 (0) = g2 (0) = 0 and x sinp x tanp−2 g10 (x) p = , 0 g2 (x) sinhp x tanhp−2 x p x tanhp−3 xh(x) tanp−3 d g10 (x) p p = 2 , 0 p−2 d x g2 (x) sinhp x tanhp x h(x) = (p − 2) secpp x − 1 tanp x + 1 − (p − 2) sechpp x tanhp x. π Owing to p ≥ 3 and x ∈ x∗p , 2p , by the monotonicity of secp x and sechp x, we obtain (p − 2) secpp x − 1 > 0

4. An open problem Finally, we pose an open problem. Open Problem 4.1. For p ∈ (1, 2], the inequality p sinp x tanp x px x + > + x x sinp x tanp x π is valid on 0, 2p .

(4.1)

Acknowledgements The first author was supported partially by the NSF of Shandong Province under grant numbers ZR2011AL001, ZR2012AQ028, and by the Science Foundation of Binzhou University under grant BZXYL1303, China. The authors appreciate the anonymous referees for their valuable comments on and helpful corrections to the original version of this paper.

g 0 (x)

and 1 − (p − 2) sechpp x > 0. So the ratio g10 (x) is strictly 2 π increasing for x ∈ x∗p , 2p . By Lemma 2.1, we see that g1 (x) ∗ πp g2 (x) is strictly increasing for x ∈ xp , 2 . On account of Lemmas 2.8 and 2.9, we obtain that the quotient f10 (x) f 0 (x) is strictly increasing. Thus, by Lemma 2.1, the 2

ratio

f1 (x) f2 (x)

is also strictly increasing.

Remark 3.5. If p = 3, we obtain ln

x

x∗3

= 0. This means

sin3 x that the function f (x) = sinh is strictly increasing 3x ln x on 0, π23 . Notice that Theorem 3.10 partially solves Conjecture 3.12 in [8].

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