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Sharp Wilker-type inequalities with applications
Journal of Inequalities and Applications volume 2014, Article number: 166 (2014)
Abstract
In this paper, we prove that the Wilker-type inequality
holds for any fixed and all if and only if or (), and the hyperbolic version of Wilker-type inequality
holds for any fixed () and all if and only if or ( or ). As applications, several new analytic inequalities are presented.
MSC:26D05, 33B10.
1 Introduction
Wilker [1] proposed two open problems, the first of which states that the inequality
holds for all . Inequality (1.1) was proved by Sumner et al. in [2].
Recently, the Wilker inequality (1.1) and its generalizations, improvements, refinements and applications have attracted the attention of many mathematicians (see [3–17] and related references therein).
In [9], Wu and Srivastava established the following Wilker-type inequality:
and its weighted and exponential generalization.
Theorem Wu ([[9], Theorem 1])
Let , and . If or , then the inequality
holds for .
As an application of inequality (1.3), an open problem was proposed, answered and improved by Sándor and Bencze in [18]. Recently, inequality (1.3) and its related inequalities in [9] were extended to Bessel functions [3], and the hyperbolic version of Theorem Wu was presented in [12].
In 2009, Zhu [16] gave another exponential generalization of Wilker inequality (1.1) as follows.
Theorem Zh1 ([[16], Theorems 1.1 and 1.2])
Let . Then the inequalities
hold if , while the first one in (1.4) holds if and only if .
Theorem Zh2 ([[16], Theorems 1.3 and 1.4])
Let . Then the inequalities
hold if , while the first one in (1.5) holds if and only if .
In [16], Zhu also proposed an open problem: find the respectively largest range of p such that inequalities (1.4) and (1.5) hold. It was solved by Matejička in [19].
Another inequality associated with the Wilker inequality is the following:
for , which is known as the Huygens inequality [20]. The following refinement of Huygens inequality is due to Neuman and Sándor [7]:
for . Very recently, the generalizations of (1.7) were given by Neuman in [8]. In [21], Zhu proved that the inequalities
hold for all with the best constants , , , . Later, Zhu [15] generalized inequalities (1.8) and (1.9) to the exponential form as follows.
Theorem Zh3 ([[15], Theorems 1.1 and 1.2])
Let . Then we have
-
(i)
If , then the double inequality
(1.10)
holds if and only if and .
-
(ii)
If , then double inequality (1.10) holds if and only if and .
-
(iii)
If , then the second inequality in (1.10) holds if and only if .
The hyperbolic version of inequalities (1.7) was given in [7] by Neuman and Sándor. Later, Zhu showed the following.
Theorem Zh4 ([[17], Theorem 4.1])
Let . Then one has
-
(i)
If , then the double inequality
(1.11)
holds if and only if and .
-
(ii)
If , then the inequality
(1.12)
holds if and only if .
The main aim of this paper is to present the best possible parameter p such that the inequalities
or their reversed inequalities hold for certain fixed k with . As applications, we also present several new analytic inequalities.
2 Lemmas
In order to establish our main results, we need several lemmas, which we present in this section.
Lemma 1 Let A, B and C be defined on by
Then, for fixed , the function is increasing on . Moreover, we have
Proof We clearly see that for because of and , and because of
by Wilker inequality (1.1).
Let , then simple computations lead to
It follows from [[16], Lemma 2.9] that the function is positive and increasing on . Hence it remains to prove that the function is also positive and increasing. Clearly, , we only need to show that for . Indeed,
which is clearly positive due to Wilker inequality (1.1). Therefore, is increasing on , and
This completes the proof. □
Lemma 2 Let E, F and G be defined on by
Then, for fixed (), the function is decreasing (increasing) on . Moreover, we have
Proof It is easy to verify that for due to
While because of
by Wilker inequality (1.5).
Denote by H and simple computations give
Clearly, , and it was proved in [[19], Proof of Lemma 2.2] that is decreasing on . In order to prove the monotonicity of H, we only need to deal with the sign and monotonicity of .
-
(i)
Clearly, for . And we claim that is also decreasing on . Indeed,
Consequently, is positive and decreasing on , and so
-
(ii)
For , by the previous proof we clearly see that is decreasing on , and so
which implies that is positive and decreasing on , and so is the function . That is, H is negative and increasing on , and inequality (2.8) holds true.
This completes the proof. □
Remark 1 It should be noted that for and for . In fact, it suffices to notice (2.8) and for .
Lemma 3 For , we have
Proof It suffices to show that
for .
Differentiation gives
for . Therefore, Lemma 3 follows from and . □
3 Main results
Theorem 1 For fixed , inequality (1.13) holds for if and only if or .
Proof Inequality (1.13) is equivalent to
for . Differentiation yields
where
A simple computation leads to .
Differentiation again and simplifying give
where
where , and are defined as in (2.1), (2.2) and (2.3), respectively.
By (3.2), (3.4) we easily get
Necessity. We first present two limit relations:
In fact, using power series extension yields
which implies the first limit relation (3.8). From the fact that , the second one (3.9) easily follows.
Now we can derive that the necessary condition of (1.13) holds for from the simultaneous inequalities and . Solving for p yields or
where the equality holds due to Lemma 3.
Sufficiency. We prove that the condition or is sufficient. We divide the proof into three cases.
Case 1 . Clearly, , then and , which together with yields and .
Case 2 . By Lemma 1 it is easy to get
which reveals that , and , which in combination with implies and .
Case 3 . Lemma 1 reveals that is increasing on , so is the function . Since
there exists such that for and for , and so is . Therefore, for but , which implies that there exists such that for and for . Due to , it is deduced that for and for , which reveals that f is increasing on and decreasing on . It follows that
that is, for .
This completes the proof. □
Theorem 2 For fixed , the reversed inequality of (1.13), that is,
holds for if and only if .
Proof Necessity. If inequality (3.10) holds for , then we have
Solving the inequality for p yields .
Sufficiency. We prove that the condition is sufficient. It suffices to show that for . By Lemma 1 it is easy to get
which reveals that , and . In combination with , it implies . Thus, , which proves the sufficiency and the proof is completed. □
Theorem 3 For fixed , inequality (1.14) holds for if and only if or .
Proof Let
Then inequality (1.14) is equivalent to . Differentiation leads to
where
Differentiation again gives
where
where , and are defined as in (2.5), (2.6) and (2.7), respectively.
By (3.12) and (3.14) we easily get
Necessity. If inequality (1.14) holds for , then we have . Expanding in power series gives
Hence we get
Solving the inequality for p yields or .
Sufficiency. We prove that the condition or is sufficient for (1.14) to hold.
If , then due to . Hence, from (3.17) we have and . It is derived by (3.16) that , and so .
If , then by Lemma 2 we have
and
From (3.17) we have and . It follows by (3.16) that , which implies that .
This completes the proof. □
Remark 2 For , since for and for , there does not exist p such that the reverse inequality of (1.14) holds for all . But we can show that there exists such that , that is, the reverse inequality of (1.14) holds for . The details of the proof are omitted.
Theorem 4 For fixed , the reverse of (1.14), that is,
holds for if and only if or .
Proof Necessity. If inequality (3.18) holds for , then we have
Solving the inequality for p yields or .
Sufficiency. We prove that the condition or is sufficient for (3.18) to hold.
If , then due to and . Hence, from (3.17) we have and . It is derived by (3.16) that , and so .
If , then by Lemma 2 we have
and
From (3.17) we have and . It follows by (3.16) that , which implies that .
This completes the proof. □
4 Applications
4.1 Huygens-type inequalities
Letting in Theorems 1 and 2, we have the following proposition.
Proposition 1 For , the double inequality
holds if and only if or and .
Let denote the r th weighted power mean of positive numbers defined by
where .
Since
by Proposition 1 the inequality
holds for if and only if . Similarly, its reversed inequality holds if and only if . The facts can be stated as a corollary.
Corollary 1 Let be defined by (4.2). Then, for , the inequalities
hold if and only if and .
Remark 3 The Cusa-Huygens inequality [20] refers to
holds for , which is equivalent to the second inequality in (1.7). As an improvement and generalization, Corollary 1 was proved in [22] by Yang. Here we provide a new proof.
Remark 4 Let and let . Then , and inequalities (4.3) can be rewritten as
where P is the first Seiffert mean [23] defined by
A and G denote the arithmetic and geometric means of a and b, respectively.
Let . Then , , and inequalities (4.3) can be rewritten as
where T is the second Seiffert mean [24] defined by
Q denotes the quadratic mean of a and b.
Obviously, by Corollary 2, the two double inequalities (4.5) (see [22]) and (4.6) hold if and only if and , (4.6) seems to be a new inequality.
In the same way, taking in Theorem 3, we get the following.
Proposition 2 For , the inequality
holds if and only if or .
Similar to Corollary 1, we have the following.
Corollary 2 Let be defined by (4.2). Then, for , the inequalities
hold if and only if and .
Remark 5 Let and . Then , , and (4.8) can be rewritten as
where L is the logarithmic means of a and b defined by
Making use of yields and , where NS is the Nueman-Sándor mean defined by
Thus, (4.8) is equivalent to
Corollary 2 implies that inequalities (4.9) and (4.10) hold if and only if and . The second inequality in (4.10) is a new inequality.
Remark 6 It should be pointed out that all inequalities involving and cosx or and coshx in this paper can be rewritten as the equivalent inequalities for bivariate means mentioned previously. In what follows we no longer mention this.
4.2 Wilker-Zhu-type inequalities
Letting in Theorems 1 and 2, we have the following.
Proposition 3 For , the double inequality
holds if and only if or and .
Note that
By Proposition 3 the inequality
or
holds for if and only if , where is defined on by
Likewise, its reversed inequality holds if and only if . This result can be stated as a corollary.
Corollary 3 Let be defined by (4.12). Then, for , the inequalities
are true if and only if and .
Taking in Theorem 3, we have the following.
Proposition 4 For , the inequality
holds if and only if or .
In a similar way, we get Corollary 4.
Corollary 4 Let be defined by (4.12). Then, for , the inequalities
are true if and only if and .
Now we give a generalization of inequalities (1.4) given by Zhu [15].
Proposition 5 For fixed , both chains of inequalities
hold for if and only if and .
Proof The first inequality in (4.15) is equivalent to
Due to and , it holds for if and only if
The second one is equivalent to
which can be simplified to
It is true for if and only if .
By Theorem 1, the third one in (4.15) holds for if and only if
Hence, inequalities (4.15) hold for if and only if
which proves (4.15).
In the same way, we can prove (4.16), the details are omitted. □
Letting in Proposition 5, we have the following.
Corollary 5 For , inequality (1.4) holds if and only if .
Similarly, using Theorem 3 we easily prove the following proposition.
Proposition 6 For fixed , the inequalities
hold for if and only if and .
Letting in Proposition 6, we have the following.
Corollary 6 For , inequality (1.5) holds if and only if .
Remark 7 Clearly, Corollaries 5 and 6 offer another method for solving the problems posed by Zhu in [16].
4.3 Other Wilker-type inequalities
Taking in Theorems 1 and 2, we obtain the following.
Proposition 7 For , the inequality
holds if and only if or . It is reversed if and only if .
Proposition 8 For , the inequality
holds if and only if or . It is reversed if and only if .
Putting in Theorem 3, we get the following.
Proposition 9 For , the inequality
holds if and only if or .
Proposition 10 For , the inequality
holds if and only if or .
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Acknowledgements
The authors would like to express their deep gratitude to the referees for giving many valuable suggestions. The research was supported by the Natural Science Foundation of China under Grants 61374086 and 11171307, and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.
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Authors’ contributions
Z-HY carried out the proof of the Wilker-type inequality and drafted the manuscript. Y-MC provided the main idea and carried out the proof of the hyperbolic version of Wilker-type inequality. All authors read and approved the final manuscript.
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Yang, ZH., Chu, YM. Sharp Wilker-type inequalities with applications. J Inequal Appl 2014, 166 (2014). https://doi.org/10.1186/1029-242X-2014-166
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DOI: https://doi.org/10.1186/1029-242X-2014-166