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Generalized Wilker-type inequalities with two parameters
Journal of Inequalities and Applications volume 2016, Article number: 187 (2016)
Abstract
In the article, we present certain \(p, q\in\mathbb{R}\) such that the Wilker-type inequalities
hold for all \(x\in(0, \pi/2)\).
1 Introduction
The well-known Wilker inequality \((\sin x/x)^{2}+\tan x/x>2\) for all \(x\in(0, \pi/2)\) was proposed by Wilker [1] and proved by Sumner et al. [2].
Recently, the Wilker inequality has attracted the attention of many researchers. Many generalizations, improvements, and refinements of the Wilker inequality can be found in the literature [3–10].
Pinelis [11] and Sun and Zhu [12] proved that the inequalities
hold for all \(x\in(0, \pi/2)\) and \(y>0\) if and only if \(\lambda\leq 8/45\) and \(\mu\geq2/45\).
Wu and Srivastava [13] provided polynomials \(P_{1}(x)\) and \(P_{2}(x)\) of degree \(2n+3\) \((n\in\mathbb{N})\) with explicit expressions and coefficients concerning Bernoulli numbers such that the double inequality
holds for all \(x\in(0, \pi/2)\).
Yang [14] proved that \(p=5/3\) and \(q=\log2/[2(\log\pi-\log2)]\) are the best possible parameters such that the double inequality
holds for all \(x\in(0, \pi/2)\).
Very recently, Yang and Chu [15] proved that the Wilker-type inequality
holds for any fixed \(k\geq1\) and all \(x\in(0, \pi/2)\) if and only if \(p>0\) or \(p\leq[\log2-\log(k+2)]/[k(\log\pi-\log2)]\) (\(-12/[5(k+2)]\leq p<0\)), and the hyperbolic version of Wilker-type inequality
holds for any fixed \(k\geq1\) (\({<}-2\)) and all \(x\in(0, \infty)\) if and only if \(p>0\) or \(p\leq-12/[5(k+2)]\) (\(p<0\) or \(p\geq -12/[5(k+2)]\)).
More results of the Wilker-type inequalities for hyperbolic, Bessel, circular, inverse trigonometric, inverse hyperbolic, lemniscate, generalized trigonometric, generalized hyperbolic, Jacobian elliptic and theta, and hyperbolic Fibonacci functions can be found in the literature [16–28].
The main purpose of the article is to establish the Wilker-type inequalities
and
for all \(x\in(0, \pi/2)\) and certain \(p, q\in\mathbb{R}\). Some complicated analytical computations are carried out using the computer algebra system Mathematica.
2 Lemmas
In order to prove our main results, we need several lemmas.
Lemma 2.1
Let \(-\infty< a< b<\infty\), \(f, g: [a, b]\rightarrow\mathbb{R}\) be continuous on \([a, b]\) and differentiable on \((a, b)\), and \(g^{\prime}(x)\neq0\) on \((a, b)\). Then both of the functions
are increasing (decreasing) on \((a, b)\) if \(f^{\prime}(x)/g^{\prime}(x)\) is increasing (decreasing) on \((a, b)\). If \(f^{\prime}(x)/g^{\prime}(x)\) is strictly monotone, then the monotonicity in the conclusion is also strict.
Lemma 2.2
(See [31])
Let \(A(t)=\sum_{k=0}^{\infty }a_{k}t^{k}\) and \(B(t)=\sum_{k=0}^{\infty}b_{k}t^{k}\) be two real power series converging on \((-r,r)\) (\(r>0\)) with \(b_{k}>0\) for all k. If the nonconstant sequence \(\{a_{k}/b_{k}\}\) is increasing (decreasing) for all k, then the function \(t\mapsto A(t)/B(t)\) is strictly increasing (decreasing) on \((0,r)\).
Lemma 2.3
(See [32])
Let \(n\in\mathbb{N}\), and \(B_{n}\) be the Bernoulli numbers. Then the power series formulas
hold for \(x\in(-\pi, \pi)\), and the power series formulas
hold for \(x\in(-\pi/2, \pi/2)\).
Lemma 2.4
(See [33])
Let \(B_{n}\) be the Bernoulli numbers. Then the double inequality
holds for all \(n\in\mathbb{N}\).
From Lemma 2.4 we immediately get the following:
Remark 2.1
Let \(B_{n}\) be the Bernoulli numbers. Then the double inequality
holds for all \(n\in\mathbb{N}\) and \(n\geq1\).
Lemma 2.5
Let \(n\in\mathbb{N}\), \(B_{n}\) be the Bernoulli numbers, and \(a_{n}\) and \(b_{n}\) be respectively defined by
Then the sequence \(\{b_{n}/a_{n}\}\) is strictly increasing for \(n\geq3\).
Proof
Let \(n\geq3\) and
Then from (2.1)-(2.3) and Remark 2.1 we get
Let
Then we clearly see that
for \(n\geq3\).
It follows from (2.6) and (2.7) that
for all \(n\geq3\).
It is not difficult to verify that
and
for all \(n\geq3\).
Therefore, Lemma 2.5 follows easily from (2.3)-(2.5) and (2.8)-(2.10). □
Lemma 2.6
Let \(n\in\mathbb{N}\), \(B_{n}\) be the Bernoulli numbers, \(u_{n}\) be defined by (2.3), and \(c_{n}\) and \(v_{n}\) be respectively defined by
Then \(v_{n}>u_{n}\) for all \(n\geq3\).
Proof
It follows from (2.1)-(2.3), (2.11), and (2.12) that
From (2.13), Remark 2.1, and the inequality \(\pi^{2}>9\) we get
Note that
and
for all \(n\geq8\).
Therefore, Lemma 2.6 follows easily from (2.14)-(2.18). □
Lemma 2.7
Let \(n\in\mathbb{N}\), and \(w_{n}\) be defined by
Then \(w_{n}>0\) for all \(n\geq5\).
Proof
Let
Then we clearly see that
for all \(n\geq5\).
Inequalities (2.20) and (2.21) lead to the conclusion that
for all \(n\geq5\).
Note that
for all \(n\geq5\).
Therefore, Lemma 2.7 follows from (2.19), (2.22), and (2.23). □
Lemma 2.8
Let \(n\in\mathbb{N}\), and \(u_{n}\) and \(v_{n}\) be defined by (2.3) and (2.12), respectively. Then \(v_{3}=37u_{3}/35\) and \(v_{n}<37u_{n}/35\) for all \(n\geq4\).
Proof
It follows from (2.1)-(2.3), (2.11), and (2.12) that
From Remark 2.1, (2.24), and the inequality \(\pi^{2}<10\) we get
where \(w_{n}\) is given in Lemma 2.7.
Therefore, Lemma 2.8 follows easily from Lemma 2.7, (2.25), and (2.26). □
Let
Then from the Wilker inequality and Lemma 2.3 we clearly see that
for all \(x\in(0, \pi/2)\) and
where \(a_{n}\), \(b_{n}\), and \(c_{n}\) are respectively given by (2.1), (2.2), and (2.11).
Lemma 2.9
Let \(q\in\mathbb{R}\), \(A(x)\), \(B(x)\), and \(C(x)\) be respectively given by (2.27)-(2.29), and \(f(x): (0, \pi/2)\rightarrow\mathbb{R}\) be defined as
Then the following statements are true:
-
(1)
if \(q=-1\), then \(f(x)\) is strictly increasing from \((0, \pi/2)\) onto \((2q+12/5, 3-\pi^{2}/4)\);
-
(2)
if \(q>-1\), then \(f(x)\) is strictly increasing from \((0, \pi/2)\) onto \((2q+12/5, \infty)\);
-
(3)
if \(q\leq-37/35\), then \(f(x)\) is strictly decreasing from \((0, \pi/2)\) onto \((-\infty, 2q+12/5)\).
Proof
Let \(a_{n}\), \(b_{n}\), \(c_{n}\), \(u_{n}\), and \(v_{n}\) be respectively defined by (2.1)-(2.3), (2.11), and (2.12). Then from (2.30)-(2.32) and Lemma 2.5 we have
for all \(n\geq3\).
Note that
We divide the proof into two cases.
Case 1 \(q\geq-1\). Then it follows from (2.34) and (2.35), together with Lemma 2.6, that
for \(n\geq3\).
Therefore, parts (1) and (2) follow from (2.33), (2.36)-(2.38), (2.40), and Lemma 2.2.
Case 2 \(q\leq-37/35\). Then (2.34) and (2.35), together with Lemma 2.8, lead to
for \(n\geq4\).
Therefore, part (3) follows from (2.33), (2.36), (2.39), (2.41), (2.42), and Lemma 2.2. □
Let \(p, q\in\mathbb{R}\), \(x\in(0, \pi/2)\), and the functions \(x\rightarrow S_{p}(x)\), \(x\rightarrow T_{q}(x)\), and \(x\rightarrow W_{p,q}(x)\) be respectively defined by
and
Then we clearly see that
Lemma 2.10
Let \(x\in(0, \pi/2)\), and \(W_{p, q}(x)\) be defined by (2.45). Then the following statements are true:
-
(1)
\(W_{p, q}(x)\) is strictly decreasing on \((0, \pi/2)\) if \(q\geq -1\) and \(p+2q+12/5\geq0\);
-
(2)
\(W_{p, q}(x)\) is strictly increasing on \((0, \pi/2)\) if \(-37/35< q\leq-1\) and \(p\leq\pi^{2}/4-3\);
-
(3)
\(W_{p, q}(x)\) is strictly increasing on \((0, \pi/2)\) if \(q\leq -37/35\) and \(p+2q+12/5\leq0\).
Proof
Let \(pq\neq0\) and \(x\in(0, \pi/2)\). Then (2.43) and (2.44) lead to
where \(A(x)\) and \(f(x)\) are respectively given by (2.27) and (2.32).
(1) If \(q\geq-1\) and \(p+2q+12/5\geq0\), then from Lemma 2.9(1) and (2) and from (2.48) we have
for \(x\in(0, \pi/2)\).
Therefore, Lemma 2.10(1) follows easily from (2.45) and (2.49) together with Lemma 2.1.
(2) If \(-37/35< q\leq-1\) and \(p\leq\pi^{2}/4-3\), then (2.48) and Lemma 2.9(1) lead to
for \(x\in(0, \pi/2)\).
Therefore, Lemma 2.10(2) follows from (2.45) and (2.50) together with Lemma 2.1.
(3) If \(q\leq-37/35\) and \(p+2q+12/5\leq0\), then Lemma 2.9(3) and (2.48) lead to the conclusion that
for \(x\in(0, \pi/2)\).
Therefore, Lemma 2.10(3) follows from (2.45) and (2.51) together with Lemma 2.1. □
Remark 2.2
It is not difficult to verify that (2.48) is also true if \(pq=0\).
3 Main results
Let
Theorem 3.1
Let \(G_{1}\), \(G_{2}\), \(G_{3}\), and \(G_{4}\) be respectively defined by (3.10)-(3.13). Then the Wilker-type inequality
holds for all \(x\in(0, \pi/2)\) if \((p, q)\in G_{1}\cup G_{2}\), and inequality (3.16) is reversed if \((p, q)\in G_{3}\cup G_{4}\).
Proof
Let \(W_{p,q}(x)\) be defined by (2.45). We only prove that inequality (3.16) holds for all \(x\in(0, \pi/2)\) if \((p, q)\in G_{1}\cup G_{2}\); the reversed inequality for \((p, q)\in G_{3}\cup G_{4}\) can be proved by a completely similar method.
We divide the proof into two cases.
Case 1 \((p, q)\in G_{1}\). Then (3.1), (3.4), and (3.6) lead to
It follows from (2.45), (2.46), Lemma 2.10(1), and (3.17) that
for \(x\in(0, \pi/2)\).
Therefore, inequality (3.16) follows easily from (3.18) and (3.19).
Case 2 \((p, q)\in G_{2}\). Then from (2.45), (2.46), Lemma 2.10(2) and (3), (3.2)-(3.4), and (3.6) we clearly see that
and
Therefore, inequality (3.16) follows from (3.20) and (3.21). □
Theorem 3.2
Let \(G_{5}\), \(G_{6}\), \(G_{7}\), and \(G_{8}\) be respectively defined by (3.11) and (3.13)-(3.15). Then the Wilker-type inequality
holds for all \(x\in(0, \pi/2)\) if \((p, q)\in G_{5}\cup G_{6}\), and inequality (3.22) is reversed if \((p, q)\in G_{7}\cup G_{8}\).
Proof
Let \(W_{p,q}(x)\) be defined by (2.45). We only prove that inequality (3.22) holds for all \(x\in(0, \pi/2)\) if \((p, q)\in G_{5}\cup G_{6}\); the reversed inequality for \((p, q)\in G_{7}\cup G_{8}\) can be proved by a completely similar method.
We divide the proof into two cases.
Case 1 \((p, q)\in G_{5}\). Then from (2.45), (2.47), Lemma 2.10(1), (3.1), (3.5), and (3.8) we clearly see that
and
Therefore, inequality (3.22) follows easily from (3.23) and (3.24).
Case 2 \((p, q)\in G_{6}\). Then (2.45), (2.47), Lemma 2.10(2) and (3), (3.2), (3.3), (3.5), and (3.8) lead to the conclusion that
and
Therefore, inequality (3.22) follows easily from (3.25) and (3.26). □
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Acknowledgements
The research was supported by the Natural Science Foundation of China under Grants 11371125, 61374086, and 11401191 and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.
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Chu, HH., Yang, ZH., Chu, YM. et al. Generalized Wilker-type inequalities with two parameters. J Inequal Appl 2016, 187 (2016). https://doi.org/10.1186/s13660-016-1127-8
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DOI: https://doi.org/10.1186/s13660-016-1127-8