Abstract
Based on the Padé approximation method, we present new inequalities for Gauss lemniscate functions. We also solve a conjecture on inequalities for Gauss lemniscate functions proposed by Sun and Chen.
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1 Introduction
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 arcsl 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
The functions (1.1) and (1.2) can be found (see [1], Chapter 1, [2], p. 259, and [3–11]).
Another pair of lemniscate functions, the arc lemniscate tangent arctl and the hyperbolic arc lemniscate tangent arctlh, have been introduced in [4], (3.1)-(3.2). Therein it has been proven that
and
(see [4], Proposition 3.1).
Recently, numerous inequalities have been given for the lemniscate functions [6, 9–11]. For example, Neuman [6] proved the following inequalities:
and
for \(0<|x|<1\).
Shafer [12] indicated several elementary quadratic approximations of selected functions without proof. Subsequently, Shafer [13] established these results as analytic inequalities. For example, Shafer [13] proved that, for \(x>0\),
The inequality (1.7) can also be found in [14]. Zhu [15] developed (1.7) to produce a symmetric double inequality
where the constants \(80/3\) and \(256/\pi^{2}\) are the best possible. In [15], (1.8) is called a Shafer-type inequality.
Mortici and Srivastava [16] presented new bounds for arctanx. Some inequalities for trigonometric functions were refined in [17].
Very recently, Sun and Chen [18] established the following Shafer-type inequalities for the lemniscate functions:
and presented the following conjecture.
Conjecture 1.1
For \(x>0\),
and
Based on the Padé approximation method, in this paper we present new inequalities for Gauss lemniscate functions. We also prove Conjecture 1.1.
Some computations in this paper were performed using Maple software.
2 Padé approximant
For later use, we introduce the Padé approximant (see [19–21]). Let f be a formal power series,
The Padé approximation of order \((p, q)\) of the function f is the rational function, denoted by
where \(p\geq0\) and \(q\geq1\) are any given integers, the coefficients \(a_{j}\) and \(b_{j}\) are given by (see [19, 21])
and we have
Thus, the first \(p + q + 1\) coefficients of the series expansion of \([p/q]_{f}\) are identical to those of f. Moreover, we have (see [20])
with \(f_{n}(x) = c_{0}+ c_{1}x+ \cdots+ c_{n}x^{n}\), the nth partial sum of the series f (\(f_{n}\) is identically zero for \(n < 0\)).
Chen [9] presented the following power-series expansions (for \(\vert x\vert <1\)):
and
We now consider the Padé approximant for the function \(\frac{\operatorname {arcsl}x}{x}\) at the point \(x=0\). Let
with the coefficients \(c_{j}\) given by
Let us find the \((2, 2)\) Padé approximant for the function (2.10) at the point \(t=0\),
Noting that
holds, we have, by (2.3),
that is,
We thus obtain
and we have, by (2.4),
That is
Replacing t by \(x^{4}\) in (2.15) yields
Remark 2.1
Using (2.5), we can also derive (2.13). Indeed, we have
Following the same method as used in the derivation of the formula (2.16), we find
and
In view of (2.16) and (2.17), we pose the following.
Conjecture 2.1
Let
and
Then the coefficients \(a_{j}\) and \(\alpha_{j}\) satisfy the following relation:
and the coefficients \(b_{j}\) and \(\beta_{j}\) satisfy the following relation:
In view of (2.19) and (2.20), we pose the following.
Conjecture 2.2
Let
and
Then the coefficients \(p_{j}\) and \(r_{j}\) satisfy the following relation:
and the coefficients \(q_{j}\) and \(s_{j}\) satisfy the following relation:
3 Inequalities
Equations (2.16)-(2.20) motivate us to establish the following theorems.
Theorem 3.1
For \(0< x<1\),
Proof
Consider the function
Differentiation yields
Elementary calculations reveal that
where
We now prove that \(f'(x)>0\) for \(0< x<1\). It suffices to show that \(g(t)>0\) for \(0< t<1\). Differentiation yields
and
We then obtain, for \(0< t<1\),
Hence, \(f'(x)>0\) for \(0< x<1\), and we have
The proof is complete. □
Remark 3.1
There is no strict comparison between the two lower bounds in (1.5) and (3.1).
Theorem 3.2
For \(x>0\),
Proof
Consider the function
Differentiation yields
Elementary calculations reveal that
where
Hence, \(F'(x)<0\) for \(x>0\), and we have
The proof is complete. □
Remark 3.2
For \(0< t<1\), we find
with
where
is a polynomial of the 16th degree, having all coefficients positive, and
for \(0< t<1\). So, \(I(t)>0\) for \(0< t<1\). We then see that the inequality (3.2) is sharper than the right side of (1.6).
Theorem 3.3
For \(x>0\),
Proof
Consider the function
Differentiation yields
We then obtain
Hence the first inequality in (3.3) holds for \(x>0\).
Consider the function
Differentiation yields
where
Elementary calculations reveal that
where
Hence, \(T'(x)<0\) for \(x>0\), and we have
Hence, the second inequality in (3.3) holds for \(x>0\). The proof is complete. □
Theorem 3.4
For \(0< x<1\),
Proof
Consider the function
Differentiation yields
where
Elementary calculations reveal that
where
Since \(R(t)>0\) for \(0< t<1\), we have \(H'(x)>0\) for \(0< x<1\). We then obtain
The proof is complete. □
4 Proof of Conjecture 1.1
Proof of (1.12)
It suffices to show by (3.2) that
i.e.,
Elementary calculations show that
where
We see from \(P_{24}(x)>0\) that (4.1) holds. The proof is complete. □
Proof of (1.13)
First of all, we prove the second inequality in (1.13). It suffices to show by the right-hand side of (3.3) that
i.e.,
Elementary calculations show that
where
We see from \(P_{40}(x)>0\) that (4.2) holds. Hence, the second inequality in (1.13) holds.
Second, we prove the first inequality in (1.13). We consider two cases.
Case 1. \(0< x<1\).
It suffices to show by the left-hand side of (3.3) that
i.e.,
Elementary calculations show that
which shows that (4.3) holds.
Case 2. \(x\geq1\).
Consider the function \(U(x)\) defined by
Differentiation yields
Noting that
holds, we obtain
We now show that \(U'(x)>0\) is also valid for \(1\leq x<2\). It suffices to show that
where
with
and
Differentiation yields
where
Hence, we have \(y'_{1}(x)>0\) for \(1\leq x<2\).
Let \(1\leq r \leq x \leq s \leq2\). Since \(y_{1}(x)\) is increasing and \(y_{2}(x)\) is decreasing for \(1\leq x\leq2\), we obtain
We divide the interval \([1, 2]\) into 100 subintervals:
By direct computation we get
Hence,
This proves \(U'(x)>0\) for \(1\leq x<2\).
We then obtain \(U'(x)>0\) for all \(x\geq1\), and we have
which shows the first inequality in (1.13) holds for \(x\geq1\). Thus, the first inequality in (1.13) holds for all \(x>0\). The proof is complete. □
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Liu, J., Chen, CP. Padé approximant related to inequalities for Gauss lemniscate functions. J Inequal Appl 2016, 320 (2016). https://doi.org/10.1186/s13660-016-1262-2
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DOI: https://doi.org/10.1186/s13660-016-1262-2