Some Inequalities for the Generalized Trigonometric and Hyperbolic Functions
1Department of Mathematics, Binzhou University, Binzhou City, Shandong Province, China
2College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China
3Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, China
4Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, China
In the paper, the authors establish some inequalities of the generalized trigono-metric 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
Turkish Journal of Analysis and Number Theory, 2014 2 (3),
Received June 21, 2014; Revised July 01, 2014; Accepted July 13, 2014Copyright © 2013 Science and Education Publishing. All Rights Reserved.
Cite this article:
- Yin, Li, Li-Guo Huang, and and Feng Qi. "Some Inequalities for the Generalized Trigonometric and Hyperbolic Functions." Turkish Journal of Analysis and Number Theory 2.3 (2014): 96-101.
- Yin, L. , Huang, L. , & Qi, A. F. (2014). Some Inequalities for the Generalized Trigonometric and Hyperbolic Functions. Turkish Journal of Analysis and Number Theory, 2(3), 96-101.
- Yin, Li, Li-Guo Huang, and and Feng Qi. "Some Inequalities for the Generalized Trigonometric and Hyperbolic Functions." Turkish Journal of Analysis and Number Theory 2, no. 3 (2014): 96-101.
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It is well known from calculus that
for For and the arc sine may be generalized as
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  and closely related references therein.
For the generalized cosine function is defined by
It is easy to see that
The generalized tangent function is defined as
From (1.8), it follows that
The generalized secant function is defined as
It follows from (1.8) and (1.9) that
The generalized cosecant function may be defined as
It is clear that
The generalized inverse hyperbolic sine function is defined by
The inverse of is called the generalized hyperbolic sine function and denoted by .
The generalized hyperbolic cosine function is defined as
It is easy to show that
The generalized hyperbolic tangent function and the generalized hyperbolic secant function are defined as
Their derivatives are
The above formulas (1.5) to (1.9) and (1.15) to (1.19) can be found in . 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
and the identities and hold. From this, it follows that
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 .
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 .
During the last decades, many authors have studied the generalized trigonometric functions introduced in . See, for example, [1, 2, 3, 4, 7] and plenty of references therein. In , some classical inequalities for generalized trigonometric and hyperbolic functions, such as Mitrinović-Adamović inequality, Huygens' inequality, and Wilker's inequality were generalized. In , 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 . In , some Turán type inequalities for generalized trigonometric and hyperbolic functions were presented. Very recently, a conjecture posed in  was verified in .
In this paper, we will establish some inequalities of the generalized trigonometric and hyperbolic functions, partially solve a conjecture in , and finally pose an open problem.
For proving our main results, we need the following lemmas.
Lemma 2.1 ([, 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 ([, Bernoulli inequality]). For and , we have
Lemma 2.3 ([8, Theorem 3.4]). For and we have
Lemma 2.4 ([, Theorem 3.16]). If , then
Lemma 2.5 ([, Theorem 3.22]). For the double inequality
holds for all
Lemma 2.6 ([, Theorem 3.24]). For all
1. if then
2. if , then
Lemma 2.7 ([8, Theorems 3.6 and 3.7]). For all we have
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
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
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
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 [, Section 8.1].
Theorem 3.2. For and we have
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
Proof. This follows from using (2.5) and
Theorem 3.4. For and we have
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 .
Theorem 3.5. For we have
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
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
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
Proof. Let An easy computation yields
Hence, is strictly increasing, and The inequality (3.9) follows.
Theorem 3.9. For and we have
Proof. Set Then
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
Theorem 3.10. For and the function is strictly increasing, where satisfies the equation
Proof. Let and Then and
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 .
Theorem 3.11. For and we have
Proof. If putting the left hand side of (3.14) becomes
Similarly, taking the right hand side of (3.14) reduces to
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.
is valid on
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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