1. Introduction

It is well known from calculus that

(1.1)

and

(1.2)

for For and the arc sine may be generalized as

(1.3)

and

(1.4)

where denotes the classical gamma function. Hence, we have

The inverse of on is called the generalized sine function, denoted by , and may be extended to See ^[7] and closely related references therein.

For the generalized cosine function is defined by

(1.5)

It is easy to see that

(1.6)

and

(1.7)

The generalized tangent function is defined as

(1.8)

From (1.8), it follows that

(1.9)

The generalized secant function is defined as

(1.10)

It follows from (1.8) and (1.9) that

(1.11)

and

(1.12)

The generalized cosecant function may be defined as

(1.13)

It is clear that

(1.14)

for

The generalized inverse hyperbolic sine function is defined by

(1.15)

The inverse of is called the generalized hyperbolic sine function and denoted by .

The generalized hyperbolic cosine function is defined as

(1.16)

It is easy to show that

(1.17)

and

(1.18)

The generalized hyperbolic tangent function and the generalized hyperbolic secant function are defined as

(1.19)

Their derivatives are

(1.20)

and

(1.21)

The above formulas (1.5) to (1.9) and (1.15) to (1.19) can be found in ^[8]. For the set is the equilateral hyperbola in the plane with the metric. The connection between the hyperbolic coordinates and the rectangular coordinates is given by and We may easily obtain that, when and are related by When the analogue of the equilateral hyperbola is the p-equilateral hyperbola

(1.22)

and the identities and hold. From this, it follows that

and

This gives a geometrical interpretation to and 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 - trigonometric functions were given. Recently, the functions and 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 Turán 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.

2. Lemmas

For proving our main results, we need the following lemmas.

Lemma 2.1 ([^[12], Lemma 2.9]). Let and be continuous on and differentiable on such that on If is increasing (or decreasing, respectively) on so are the functions and

Lemma 2.2 ([^[10], Bernoulli inequality]). For and , we have

(2.1)

Lemma 2.3 ([8, Theorem 3.4]). For and we have

(2.2)

Lemma 2.4 ([^[8], Theorem 3.16]). If , then

(2.3)

and

(2.4)

Lemma 2.5 ([^[8], Theorem 3.22]). For the double inequality

(2.5)

holds for all

Lemma 2.6 ([^[8], Theorem 3.24]). For all

1. if then

(2.6)

2. if , then

(2.7)

Lemma 2.7 ([8, Theorems 3.6 and 3.7]). For all we have

(2.8)

and

(2.9)

where the constants and are the best possible.

Lemma 2.8. For and the function is positive and strictly increasing.

Proof. It is apparent that the function is positive.

An easy computation yields and

where

This means that is strictly increasing and Hence, it follows that and that is strictly increasing.

Lemma 2.9. For and the functions and are positive.

Proof. An easy computation yields

and

These imply that and that

3. Main Results

Now we are in a position to present our main results.

Theorem 3.1. For and we have

(3.1)

Proof. Setting and in (2.1) leads to

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

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].

Theorem 3.2. For and we have

(3.2)

Proof. Letting and in (2.1) gives

Combining this with (2.2) and (2.3) yields (3.2).

Theorem 3.3. For and we have

(3.3)

Proof. This follows from using (2.5) and

Theorem 3.4. For and we have

(3.4)

Proof. This follows from using (2.6) and

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].

Theorem 3.5. For we have

(3.5)

and

(3.6)

Proof. Taking and in (2.1) and using the inequality (3.3) result in

The inequality (3.6) may be deduced similarly.

Theorem 3.6. For and we have

(3.7)

Proof. When letting and the inequality (3.7) becomes which is equivalent to , that is,

This can be derived from utilizing (2.8) as follows

Theorem 3.7. For and we have

(3.8)

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

Remark 3.3. Inequalities (3.7) and (3.8) imply inequalities (3.1) and (3.20) in [8, Corollary 3.19].

Theorem 3.8. For and we have

(3.9)

Proof. Let An easy computation yields

Hence, is strictly increasing, and The inequality (3.9) follows.

Theorem 3.9. For and we have

(3.10)

or, equivalently,

(3.11)

Proof. Set Then

and

Accordingly, it follows that is strictly decreasing, and The inequalities (3.10) and (3.11) are proved.

Remark 3.4. Considering Theorem 3.9 and reveals

(3.12)

Theorem 3.10. For and the function is strictly increasing, where satisfies the equation

Proof. Let and Then and

(3.13)

where and satisfy and

Owing to and by the monotonicity of and we obtain and So the ratio is strictly increasing for By Lemma 2.1, we see that is strictly increasing for On account of Lemmas 2.8 and 2.9, we obtain that the quotient is strictly increasing. Thus, by Lemma 2.1, the ratio is also strictly increasing.

Remark 3.5. If , we obtain This means that the function is strictly increasing on

Notice that Theorem 3.10 partially solves Conjecture 3.12 in ^[8].

Theorem 3.11. For and we have

(3.14)

Proof. If putting the left hand side of (3.14) becomes

(3.15)

Similarly, taking the right hand side of (3.14) reduces to

(3.16)

Making use of the monotonicity of and acquires

The inequality (3.14) is thus proved.

4. An Open Problem

Finally, we pose an open problem.

(4.1)

is valid on

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.

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