Refinements and Sharpening of some Huygens and Wilker Type Inequalities
1Department of Information Engineering, Weihai Vocational University, Weihai, Shandong, China
2Department of Mathematics, Chongqing Normal University, Chongqing City, China
3College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China
4Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, China
5Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, China
In the article, some Huygens and Wilker type inequalities involving trigonometric and hyperbolic functions are refined and sharpened.
Keywords: refinement, sharpening, Huygens inequality, Wilker inequality, trigonometric function, hyperbolic function
Turkish Journal of Analysis and Number Theory, 2014 2 (4),
Received June 30, 2014; Revised August 16, 2014; Accepted August 21, 2014Copyright © 2014 Science and Education Publishing. All Rights Reserved.
Cite this article:
- Jiang, Wei-Dong, Qiu-Ming Luo, and Feng Qi. "Refinements and Sharpening of some Huygens and Wilker Type Inequalities." Turkish Journal of Analysis and Number Theory 2.4 (2014): 134-139.
- Jiang, W. , Luo, Q. , & Qi, F. (2014). Refinements and Sharpening of some Huygens and Wilker Type Inequalities. Turkish Journal of Analysis and Number Theory, 2(4), 134-139.
- Jiang, Wei-Dong, Qiu-Ming Luo, and Feng Qi. "Refinements and Sharpening of some Huygens and Wilker Type Inequalities." Turkish Journal of Analysis and Number Theory 2, no. 4 (2014): 134-139.
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The famous Huygens inequality for the sine and tangent functions states that for
The hyperbolic counterpart of (1.1) was established in  as follows: For
The inequalities (1.1) and (1.2) were respectively refined in [, Theorem 2.6] as
In  the inequality (1.2) was improved as
In , the following inequality is given
In , Wilker proved
and proposed that there exists a largest constant c such that
for In , the best constant c in (1.7) was found and it was proved that
Recently the inequalities (1.3) and (1.6) were respectively refined in  as
The hyperbolic counterparts of the last two inequalities in (1.8) were also given in  as follows:
The aim of this paper is to refine and sharpen some of the above-mentioned Huygens and Wilker type inequalities.
2. Some Lemmas
In order to attain our aim, we need several lemmas below.
Lemma 2.1. The Bernoulli numbers for have the property
where the Bernoulli numbers for are defined by
Proof. In [, p. 16 and p. 56], it is listed that for
where is the Riemann zeta function defined by
From (2.2), the formula (2.1) follows.
Lemma 2.2. For we have
Proof. This is an easy consequence of combining the equality
see [, p. 75, 4.3.68], with Lemma 2.1.
Lemma 2.3 ([, p. 75, 4.3.70]). For
Lemma 2.4. For
the formula (2.7) follows from differentiating (2.6).
Lemma 2.5. For
Proof. This follows from differentiating on both sides of (2.5) and using (2.1).
Lemma 2.6. For
with Lemma 2.5, the identity (2.5), and Lemma 2.1 gives (2.9).
The equality (2.10) follows from combination of
with Lemma 2.4.
Lemma 2.7. Let f and g be continuous on and differentiable in such that in If is increasing (or decreasing) in , then the functions and are also increasing (or decreasing) in
Lemma 2.8. Let and for be real numbers and the power series
be convergent on for some If and the ratio is strictly increasing for then the function is also strictly increasing on
3. Main Results
Now we are in a position to state and prove our main results, refinements and sharpening of some Huygens and Wilker type inequalities mentioned in the first section.
Theorem 3.1. For we have
The scalars and in (3.1) are the best possible.
for . By virtue of (2.4), (2.7), and (2.8), we
So the function is strictly increasing on . Moreover, it is easy to obtain
The proof of Theorem 3.1 is complete.
Theorem 3.2. For
The constants and in (3.2) are the best possible.
Proof. By using (2.4), (2.7), (2.9), and (2.10), the function
may be expanded as
for the sequence is increasing and Thus, the function is increasing on Moreover,
The proof of Theorem 3.2 is complete.
Theorem 3.3. For x > 0, we have
The constant is the best possible.
From the power series expansions
it follows that
It is easy to see that the quotient
for This means that the sequence is increasing. By Lemma 2.8, the function is increasing on , and so, by Lemma 2.7, the function is increasing on Moreover, it is not difficult to obtain Theorem 3.3 is thus proved.
Theorem 3.4. For x > 0,
The number in (3.6) is the best possible.
By the power series expansions in (3.4), we obtain
Furthermore, when by a simple computation, we have
for . Hence, the sequence is increasing. By Lemma 2.8, the function is increasing. Finally, it is easy to see that The proof of Theorem 3.4 is complete.
Theorem 3.5. For we have
The numbers and are the best possible.
By (2.4), (2.6), and (2.7), we have
This shows that the function is increasing on . Moreover, it is straightforward to obtain
The proof of Theorem 3.5 is complete.
Remark 3.1. This paper is a slightly revised version of the preprint .
The first author was partially supported by the Project of Shandong Province Higher Educational Science and Technology Program under grant No. J11LA57, China. The second author was partially supported by the NNSF under Grant No. 11361038, by the Natural Science Foundation Project of Chongqing City under Grant No. CSTC2011JJA00024, by the Research Project of Science and Technology of Chongqing Education Commission under Grant No. KJ120625, and by the Fund of Chongqing Normal University, China under Grant No. 10XLR017 and 2011XLZ07, China
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