1. Introduction
The famous Huygens inequality for the sine and tangent functions states that for
 (1.1) 
The hyperbolic counterpart of (1.1) was established in ^{[10]} as follows: For
 (1.2) 
The inequalities (1.1) and (1.2) were respectively refined in [^{[10]}, Theorem 2.6] as
 (1.3) 
for and
In ^{[8]} the inequality (1.2) was improved as
 (1.4) 
In ^{[10]}, the following inequality is given
 (1.5) 
For more information in this area, please refer to ^{[13, 16]}, [^{[14]}, Section 1.7 and Section 7.3] and closely related references therein.
In ^{[18]}, Wilker proved
 (1.6) 
and proposed that there exists a largest constant c such that
 (1.7) 
for In ^{[17]}, the best constant c in (1.7) was found and it was proved that
for The constants and in the above inequality are the best possible. For more information on this topic, please see ^{[4, 5, 19]}, [^{[14]}, pp. 3840, Section 8] and closely related references therein.
Recently the inequalities (1.3) and (1.6) were respectively refined in ^{[9]} as
and
 (1.8) 
The hyperbolic counterparts of the last two inequalities in (1.8) were also given in ^{[9]} as follows:
The aim of this paper is to refine and sharpen some of the abovementioned 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
 (2.1) 
where the Bernoulli numbers for are defined by
Proof. In [^{[2]}, p. 16 and p. 56], it is listed that for
 (2.2) 
where is the Riemann zeta function defined by
 (2.3) 
From (2.2), the formula (2.1) follows.
Lemma 2.2. For we have
 (2.4) 
Proof. This is an easy consequence of combining the equality
 (2.5) 
see [^{[1]}, p. 75, 4.3.68], with Lemma 2.1.
Lemma 2.3 ([^{[1]}, p. 75, 4.3.70]). For
 (2.6) 
Lemma 2.4. For
 (2.7) 
Proof. Since
the formula (2.7) follows from differentiating (2.6).
Lemma 2.5. For
 (2.8) 
Proof. This follows from differentiating on both sides of (2.5) and using (2.1).
Lemma 2.6. For
 (2.9) 
and
 (2.10) 
Proof. Combining
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
The above Lemma 2.7 can be found, for examples, in [^{[3]}, p. 292, Lemma 1], [^{[6]}, p. 57, Lemma 2.3], [^{[11]}, p. 92, Lemma 1], and [^{[12]}, p. 161, Lemma 2.3].
Lemma 2.8. Let and for be real numbers and the power series
 (2.11) 
be convergent on for some If and the ratio is strictly increasing for then the function is also strictly increasing on
The above Lemma 2.8 can be found, for examples, in [^{[3]}, p. 292, Lemma 2], [^{[15]}, p. 71, Lemma 1], and [^{[20]}, Lemma 2.2].
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
 (3.1) 
The scalars and in (3.1) are the best possible.
Proof. Let
for . By virtue of (2.4), (2.7), and (2.8), we
have
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
 (3.2) 
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
Since and
for the sequence is increasing and Thus, the function is increasing on Moreover,
and
The proof of Theorem 3.2 is complete.
Theorem 3.3. For x > 0, we have
 (3.3) 
The constant is the best possible.
Proof. Let
and let
and
From the power series expansions
 (3.4) 
and
 (3.5) 
it follows that
and
It is easy to see that the quotient
satisfies
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,
 (3.6) 
The number in (3.6) is the best possible.
Proof. Let
and let
and
By the power series expansions in (3.4), we obtain
and
The ratio
satisfies
and
Furthermore, when by a simple computation, we have
where
and
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
 (3.7) 
The numbers and are the best possible.
Proof. Let
By (2.4), (2.6), and (2.7), we have
This shows that the function is increasing on . Moreover, it is straightforward to obtain
and
The proof of Theorem 3.5 is complete.
Remark 3.1. This paper is a slightly revised version of the preprint ^{[7]}.
Acknowledgements
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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