Abstract
We establish an inequality by quadratic estimations; the double inequality
holds for \(x>0\), where the constants \((\pi^{2} -4)^{2}\) and 32 are the best possible.
Similar content being viewed by others
1 Introduction
Shafer [1–3] showed that the inequality
holds for \(x>0\). Various Shafer-type inequalities are known, and they have been applied, extended and refined, see [4–8] and [9–12]. Especially, Zhu [12] showed an upper bound for inequality (1.1) and proved that the following double inequality
holds for \(x>0\), where the constants \(80/3\) and \(256/\pi^{2}\) are the best possible. Recently, in [8], Sun and Chen proved that the following inequality
holds for \(x>0\); moreover, they showed that the inequality
holds for \(0< x< x_{0} \cong 1.4243\). In this paper, we shall establish the refinements of inequalities (1.2) and (1.3).
2 Results and discussion
Motivated by (1.2), (1.3) and (1.4), in this paper, we give inequalities involving arctangent. The following are our main results.
Theorem 2.1
For \(x>0\), we have
where the constants \((\pi^{2} -4)^{2}\) and 32 are the best possible.
Theorem 2.2
For \(x> \alpha \), we have
where the constant \(\alpha = \sqrt{\frac{9600 -1860 \pi^{2} +90 \pi^{4}}{2304 -480 \pi^{2} +25 \pi^{4}}} \cong 2.26883\) is the best possible.
Theorem 2.3
For \(x> \beta\), we have
where the constant \(\beta =\sqrt{ \frac{4096 +1536 \pi^{2} -528 \pi^{4} +24 \pi^{6} +\pi^{8}}{4096 \pi^{2} -768 \pi^{4} +36 \pi^{6}}} \cong 1.30697\) is the best possible.
Theorem 2.4
For \(x> \gamma\), we have
where the constant \(\gamma \cong 1.38918\) is the best possible and satisfies the equation
From Theorems 2.1, 2.2, 2.3 and 2.4, we can get the following proposition, immediately.
Proposition 2.5
The double inequality (2.1) is sharper than (1.2) for \(x > \alpha\). Moreover, the right-hand side of (2.1) is sharper than (1.3) for \(x>\gamma\).
2.1 Proof of Theorem 2.1
Becker-Stark’s inequality is known as the inequality
which holds for \(0< x < \pi/2\). Also, Becker-Stark’s inequality (2.5) has various applications, extensions and refinements, see [13–16] and [17–19]. Especially, Zhu [19] gave the following refinement of (2.5): The inequality
holds for \(0< x<\pi/2\), where the constants \(\lambda = (\pi^{2} -9)/(6\pi^{4})\) and \(\mu = (10 -\pi^{2} )/\pi^{4}\) are the best possible. In this paper, the result of Zhu (2.6) plays an important role in the proof of Theorem 2.1.
Proof of Theorem 2.1
The equation
is equivalent to
We set \(t= \arctan{x}\), then
First, we assume that \(0< t\leq 1/2\). Here, the derivative of \(F_{1}(t)\) is
Since we have
and
for \(0 < t <\pi/2\), the following inequality holds:
where \(F_{3}(t) = 82944000-8294400 \pi^{2} -72990720 t^{2} +7084800 \pi^{2} t^{2} +24883200 t^{4} -2246400 \pi^{2} t^{4} -4832640 t^{6} +371520 \pi^{2} t^{6} +596736 t^{8} -35904 \pi^{2} t^{8} -48192 t^{10} +2076 \pi^{2} t^{10} +2472 t^{12} -68 \pi^{2} t^{12} -74 t^{14} +\pi^{2} t^{14} +t^{16}\). We set \(s=t^{2}\), then
We shall show that the functions \(F_{4}(s)>0\), \(F_{5}(s)>0\) and \(F_{6}(s)>0\). Here,
The derivative of \(F_{7}(t)\) is
Since \(F_{7}(s)\) is strictly decreasing for \(0< s< 1/4\) and \(F_{7}(1/4) = 2077/16\), we have \(F_{4}(s) >0\).
and
Therefore, we can get \(F_{3}(t)>0\). By \(120-20 t^{2}+t^{4}>0\), \(24-12 t^{2}+t^{4}>0\) and \(-720+360 t^{2}-30 t^{4}+t^{6}<0\), thus \(F_{2}(t) <0\) and \(F_{1}(t)\) is strictly decreasing for \(0 < t <1/2\). From \(F_{1}(0+) = (\pi ^{2} -4 )^{2}-16\), we can get
for \(0< t \leq 1/2\). Next, we assume that \(1/2 < t < \pi/2\). From inequality (2.6), we have
and
where
and
We set \(s=t^{2}\), then
and
The derivatives of \(G_{3}(s)\) are
and
From the inequality
\(G''_{3}(s)<0\) and \(G'_{3}(s)\) is strictly decreasing for \(1/4 < s < \pi^{2}/4\). Since \(G'_{3}(1/4) = 12(-81 +324 \pi^{2}-1574 \pi^{4} +164 \pi^{6} -\pi^{8}) \cong -24310.3\), \(G'_{3}(s)<0\) and \(G_{3}(s)\) is strictly decreasing for \(1/4 < s < \pi^{2}/4\). Therefore, we have \(G_{1}(t) > G_{3}(\pi^{2}/4) =576 \pi^{6}\) for \(1/2< t< \pi/2\). Next, the derivatives of \(G_{4}(s)\) are
and
From the inequality
\(G''_{4}(s)<0\) and \(G'_{4}(s)\) is strictly decreasing for \(1/4 < s < \pi^{2}/4\). Since \(G'_{4}(1/4) = 4(-900+1740 \pi ^{2}-501 \pi ^{4}+122 \pi ^{6}-9 \pi ^{8}) \cong -2544.56\), \(G'_{4}(s)<0\) and \(G_{4}(s)\) is strictly decreasing for \(1/4 < s < \pi^{2}/4\). Therefore, we have \(G_{2}(t) > G_{4}(\pi^{2}/4) =48 \pi^{6}\) for \(1/2< t< \pi/2\). By the squeeze theorem, \(F_{1}(t) >16\) for \(1/2 < t< \pi/2\). Also, we have
for \(1/2 < t <\pi/2\) and
By \(300-840 \pi^{2} +327 \pi^{4} +163 \pi^{6} -19 \pi^{8} \cong 286.654\), we have
Thus, we can get \(16 < F_{1}(t) < F_{1}(0+)\) for \(0< t<\pi/2\). The proof of Theorem 2.1 is complete. □
2.2 Proof of Theorem 2.2
Proof of Theorem 2.2
We have
The derivative of \(F_{2}(x)\) is
Here, we have \(15 (16-8 \pi ^{2}+\pi ^{4}+4 \pi ^{2} x^{2} ) - 36(15+16 x^{2} ) = 3 (-100-40 \pi ^{2}+5 \pi ^{4}-192 x^{2}+20 \pi ^{2} x^{2} )\). Since \(-192 +20 \pi^{2} >0\) and \(-100-40 \pi ^{2}+5 \pi ^{4}-192 x^{2}+20 \pi ^{2} x^{2}=0\) for \(x=\sqrt{\frac{100+40 \pi ^{2}-5 \pi ^{4}}{20 \pi ^{2}-192}} \cong 1.198\), we have \(F_{3}(x)<0\) for \(0< x<\sqrt{\frac{100+40 \pi ^{2}-5 \pi ^{4}}{20 \pi ^{2}-192}}\) and \(F_{3}(x)>0\) for \(x>\sqrt{\frac{100+40 \pi ^{2}-5 \pi ^{4}}{20 \pi ^{2}-192}}\). Therefore, \(F_{2}(x)\) is strictly decreasing for \(0 < x < \sqrt{\frac{100+40 \pi ^{2}-5 \pi ^{4}}{20 \pi ^{2}-192}}\) and strictly increasing for \(x > \sqrt{\frac{100+40 \pi ^{2}-5 \pi ^{4}}{20 \pi ^{2}-192}}\). From \(F_{2}(0+)=0\) and
we can get \(F_{2}(x)>0\) for \(x> \alpha\) and α is the best possible. The proof of Theorem 2.2 is complete. □
2.3 Proof of Theorem 2.3
Proof of Theorem 2.3
We have
The derivative of \(F_{2}(x)\) is
Since \(\pi^{2} (25 \pi^{2} +256 x^{2}) - {16}^{2} (8+\pi^{2} x^{2}) = -2048 +25 \pi^{4} \cong 387.227\), we can get \(\pi^{2} (25 \pi ^{2}+256 x^{2}) > {16}^{2} (8+\pi ^{2} x^{2}) \) for \(x>0\). Therefore, \(F_{3}(x)>0\) and \(F_{2}'(x)>0\) for \(x>0\). Since \(F_{2}(x)\) is strictly increasing for \(x>0\) and
we can get \(F_{2}(x)>0\) for \(x> \beta\) and β is the best possible. The proof of Theorem 2.3 is complete. □
2.4 Proof of Theorem 2.4
Lemma 2.6
For \(x>0\), we have
Proof
We have
where \(F_{1}(x)= 37209375+357210000 x^{2}-37209375 \pi ^{2} x^{2}+567000 x^{6}+604800 x^{8}+240 x^{12}+256 x^{14}\). Here, we have
We set \(t=x^{2}\) and \(F_{2}(t) = 525+5040 t -525 \pi^{2} t +8 t^{3}\), then the derivative of \(F_{2}(t)\) is \(F'_{2}(t) = 5040-525 \pi^{2} +24 t^{2}\). Since \(F'_{2}(t)=0\) for \(t= \frac{1}{2} \sqrt{\frac{1}{2} (-1680+175 \pi ^{2} )} \cong 2.4285\), we have \(F'_{2}(t)<0\) for \(0< t< \frac{1}{2} \sqrt{\frac{1}{2} (-1680+175 \pi ^{2} )}\) and \(F'_{2}(t)>0\) for \(t>\frac{1}{2} \sqrt{\frac{1}{2} (-1680+175 \pi ^{2} )}\). Hence,
for \(t>0\). Therefore, \(F_{1}(x) >0\) and the proof of Lemma 2.6 is complete. □
Proof of Theorem 2.4
We have
where \(F_{2}(x) =151200-14175 \pi^{2} +128 x^{6} -1575 \sqrt{15} \pi^{2} \sqrt{15+16 x^{2}} +75600 \sqrt{8 +\pi^{2} x^{2}} +64 x^{6} \sqrt{8 +\pi^{2} x^{2}}\). The derivative of \(F_{2}(x)\) is
By Lemma 2.6, we have \(F'_{2}(x)>0\) and \(F_{2}(x)\) is strictly increasing for \(x>0\). From \(F_{2}(0+)=37800 (4 +4 \sqrt{2} -\pi^{2} ) \cong -8041.96\), \(F_{2}(\gamma)=0\) and \(F(\infty)=\infty\), we can get \(F_{2}(x)>0\) for \(x>\gamma\). The proof of Theorem 2.4 is complete. □
References
Shafer, RE: On quadratic approximation. SIAM J. Numer. Anal. 11(2), 447-460 (1974)
Shafer, RE: Analytic inequalities obtained by quadratic approximation. Publ. Elektroteh. Fak. Univ. Beogr., Ser. Mat. Fiz. 577-598, 96-97 (1977)
Shafer, RE: On quadratic approximation, II. Publ. Elektroteh. Fak. Univ. Beogr., Ser. Mat. Fiz. 602-633, 163-170 (1978)
Guo, B-N, Luo, Q-M, Qi, F: Sharpening and generalizations of Shafer-Fink’s double inequality for the arc sine function. Filomat 27(2), 261-265 (2013)
Malešević, BJ: Application of λ-method on Shafer-Fink’s inequality. Publ. Elektroteh. Fak. Univ. Beogr., Mat. 8, 103-105 (1997)
Malešević, BJ: An application of λ-method on inequalities of Shafer-Fink’s type. Math. Inequal. Appl. 10(3), 529-534 (2007)
Nishizawa, Y: Sharpening of Jordan’s type and Shafer-Fink’s type inequalities with exponential approximations. Appl. Math. Comput. 269, 146-154 (2015)
Sun, J-L, Chen, C-P: Shafer-type inequalities for inverse trigonometric functions and Gauss lemniscate functions. J. Inequal. Appl. 2016, 212 (2016)
Zhu, L: On Shafer-Fink inequalities. Math. Inequal. Appl. 8(4), 571-574 (2005)
Zhu, L: On Shafer-Fink-type inequality. J. Inequal. Appl. 2007, 67430 (2007)
Zhu, L: New inequalities of Shafer-Fink type for arc hyperbolic sine. J. Inequal. Appl. 2008, 368275 (2008)
Zhu, L: On a quadratic estimate of Shafer. J. Math. Inequal. 2(4), 571-574 (2008)
Chen, C-P, Cheung, W-S: Sharp Cusa and Becker-Stark inequalities. J. Inequal. Appl. 2011, 136 (2011)
Chen, C-P, Sándor, J: Sharp inequalities for trigonometric and hyperbolic functions. J. Math. Inequal. 9(1), 203-217 (2015)
Debnath, L, Mortici, C, Zhu, L: Refinements of Jordan-Steckin and Becker-Stark inequalities. Results Math. 67(1-2), 207-215 (2015)
Sun, Z-J, Zhu, L: Simple proofs of the Cusa-Huygens-type and Becker-Stark-type inequalities. J. Math. Inequal. 7(4), 563-567 (2013)
Zhu, L: Sharp Becker-Stark-type inequalities for Bessel functions. J. Inequal. Appl. 2010, 838740 (2010)
Zhu, L, Hua, J: Sharpening the Becker-Stark inequalities. J. Inequal. Appl. 2010, 931275 (2010)
Zhu, L: A refinement of the Becker-Stark inequalities. Math. Notes 93(3-4), 421-425 (2013)
Acknowledgements
I would like to thank referees for their careful reading of the manuscript and for their remarks and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that he has no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Nishizawa, Y. Refined quadratic estimations of Shafer’s inequality. J Inequal Appl 2017, 40 (2017). https://doi.org/10.1186/s13660-017-1312-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-017-1312-4