Abstract
In this paper, we present Shafer-type inequalities for inverse trigonometric functions and Gauss lemniscate functions.
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1 Introduction
Shafer [1] indicated several elementary quadratic approximations of selected functions without proof. Subsequently, Shafer [2] established these results as analytic inequalities. For example, Shafer [2] proved that for \(x>0\),
The inequality (1.1) can also be found in [3]. Also in [2], Shafer proved that for \(0< x<1\),
Zhu [4] proved that the function
is strictly decreasing for \(x>0\), and
From this one derives the following double inequality:
The constants \(80/3\) and \(256/\pi^{2}\) are the best possible. In [4], (1.3) is called Shafer-type inequality.
Using the Maple software, we find that
This fact motivated us to present a new upper bound for arctanx, which is the first aim of the present paper.
Theorem 1.1
For \(x>0\),
The second aim of the present paper is to develop (1.2) to produce a symmetric double inequality.
Theorem 1.2
For \(0< x<1\), we have
with the best possible constants
Recently, some famous inequalities for trigonometric and inverse trigonometric functions have been improved (see, for example, [5–8]).
The lemniscate, also called the lemniscate of Bernoulli, is the locus of points \((x, y)\) in the plane satisfying the equation \((x^{2} + y^{2})^{2} = x^{2} + y^{2}\). In polar coordinates \((r, \theta)\), the equation becomes \(r^{2} = \cos(2\theta)\) and its arc length is given by the function
where arcslx is called the arc lemniscate sine function studied by Gauss in 1797-1798. Another lemniscate function investigated by Gauss is the hyperbolic arc lemniscate sine function, defined as
Functions (1.7) and (1.8) can be found (see [9], Chapter 1, [10], p.259 and [11–19]).
Another pair of lemniscate functions, the arc lemniscate tangent arctl and the hyperbolic arc lemniscate tangent arctlh, have been introduced in [12], (3.1)-(3.2). Therein it has been proven that
and
(see [12], Proposition 3.1).
In analogy with (1.1), we here establish Shafer-type inequalities for the lemniscate functions, which is the last aim of the present paper.
Theorem 1.3
For \(0< x<1\),
and
Theorem 1.4
For \(x>0\),
We present the following conjecture.
Conjecture 1.1
For \(x>0\),
and
2 Lemmas
The following lemmas have been proved in [17].
Lemma 2.1
For \(|x|<1\),
Lemma 2.2
For \(0< x<1\),
3 Proofs of Theorems 1.1 to 1.4
Proof of Theorem 1.1
The inequality (1.11) is obtained by considering the function \(f(x)\) defined by
Differentiation yields
where
We now show that
By an elementary change of variable
the inequality (3.1) is equivalent to
which is true. Hence, we have
So, \(f(x)\) is strictly decreasing for \(x>0\), and we have
The proof is complete. □
Remark 3.1
Let \(x_{0}=1.4243\ldots\) . Then we have
This shows that for \(0< x< x_{0}\), the upper bound in (1.11) is better than the upper bound in (1.3). In fact, for \(x\to0\), we have
and
Proof of Theorem 1.2
The double inequality (1.11) can be written for \(0< x<1\) as
where
By an elementary change of variable,
we have
We now prove that \(F(x)\) is strictly decreasing for \(0< x<1\). It suffices to show that \(G(t)\) is strictly decreasing for \(0< t<\pi/2\). Differentiation yields
where
Elementary calculations reveal that for \(0< t<\pi/2\) and \(n\geq6\),
and
We then obtain, for \(0< t<\pi/2\) and \(n\geq6\),
Hence, for every \(t\in(0,\pi/2)\), the sequence \(n\longmapsto u_{n}(t)\) is strictly decreasing for \(n\geq6\). We then obtain from (3.2)
which implies \(G'(t)<0\) for \(0< t<\pi/2\). Hence, \(G(t)\) is strictly decreasing for \(0< t<\pi/2\), and \(F(x)\) is strictly decreasing for \(0< x<1\). So, we have
for all \(x\in(0, 1)\), with the constants \(5/3\) and \((256-25\pi^{2})/\pi^{2}\) being best possible. The proof is complete. □
Proof of Theorem 1.3
By (2.1), we find that for \(0< x<1\),
Noting that
we obtain, for \(0< x<1\),
which implies (1.11).
By (2.2), we find that for \(0< x<1\),
where
Noting that for \(0< t<1\),
and
we obtain \(A(x)>0\) for \(0< x<1\). From (3.3), we obtain (1.12). The proof is complete. □
Proof of Theorem 1.4
The inequality (1.13) is obtained by considering the function \(h(x)\) defined by
Differentiation yields
By an elementary change of variable
we have
where
We now prove that
It suffices to show that
Differentiation yields
and
Thus, we have, for \(t>15\),
Hence, \(h'(x)>0\) holds for \(x>0\), and we have
The proof is complete. □
References
Shafer, RE: On quadratic approximation. SIAM J. Numer. Anal. 11, 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)
Zhu, L: On a quadratic estimate of Shafer. J. Math. Inequal. 2, 571-574 (2008)
Mortici, C: A subtly analysis of Wilker inequality. Appl. Math. Comput. 231, 516-520 (2014)
Mortici, C, Debnath, L, Zhu, L: Refinements of Jordan-Steckin and Becker-Stark inequalities. Results Math. 67, 207-215 (2015)
Mortici, C, Srivastava, HM: Estimates for the arctangent function related to Shafer’s inequality. Colloq. Math. 136, 263-270 (2014)
Nenezić, M, Malesević, B, Mortici, C: New approximations of some expressions involving trigonometric functions. Appl. Math. Comput. 283, 299-315 (2016)
Siegel, CL: Topics in Complex Function Theory, vol. 1. Wiley, New York (1969)
Borwein, JM, Borwein, PB: Pi and the AGM: A Study in the Analytic Number Theory and Computational Complexity. Wiley, New York (1987)
Carlson, BC: Algorithms involving arithmetic and geometric means. Am. Math. Mon. 78, 496-505 (1971)
Neuman, E: On Gauss lemniscate functions and lemniscatic mean. Math. Pannon. 18, 77-94 (2007)
Neuman, E: Two-sided inequalities for the lemniscate functions. J. Inequal. Spec. Funct. 1, 1-7 (2010)
Neuman, E: On Gauss lemniscate functions and lemniscatic mean II. Math. Pannon. 23, 65-73 (2012)
Neuman, E: Inequalities for Jacobian elliptic functions and Gauss lemniscate functions. Appl. Math. Comput. 218, 7774-7782 (2012)
Neuman, E: On lemniscate functions. Integral Transforms Spec. Funct. 24, 164-171 (2013)
Chen, CP: Wilker and Huygens type inequalities for the lemniscate functions. J. Math. Inequal. 6, 673-684 (2012)
Chen, CP: Wilker and Huygens type inequalities for the lemniscate functions, II. Math. Inequal. Appl. 16, 577-586 (2013)
Deng, JE, Chen, CP: Sharp Shafer-Fink type inequalities for Gauss lemniscate functions. J. Inequal. Appl. 2014, 35 (2014)
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Sun, JL., Chen, CP. Shafer-type inequalities for inverse trigonometric functions and Gauss lemniscate functions. J Inequal Appl 2016, 212 (2016). https://doi.org/10.1186/s13660-016-1157-2
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DOI: https://doi.org/10.1186/s13660-016-1157-2