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Some Padé approximations and inequalities for the complete elliptic integrals of the first kind
Journal of Inequalities and Applications volume 2021, Article number: 37 (2021)
Abstract
In this paper, we present Padé approximations of some functions involving complete elliptic integrals of the first kind \(K(x)\), and motivated by these approximations we also present the following double inequality:
Our results have superiority over some new recent results.
1 Introduction
It is well known that the complete elliptic integrals of the first kind and of the second kind are classical integrals, and apart from their theoretical importance in the theory of theta functions, they have important applications in mechanics, statistical mechanics, electrodynamics, magnetic field calculations, astronomy, geodesy, quasiconformal mappings, and other fields of mathematics and mathematical physics. In most applications, we encounter complicated expressions involving the complete elliptic integrals (which are not always in a form that is immediately recognizable), and it is difficult to find numerical values of such expressions to a sufficient number of significant digits. The complete elliptic integrals cannot be expressed in terms of elementary functions and have representations as infinite series that slowly converge, so these series are not the most computationally efficient approach for most scientists and engineers. Therefore, there is a need for appropriate approximations and bounds for these integrals.
The complete elliptic integrals of the first and second kinds \(K(x)\) and \(E(x)\), respectively, are defined as [9, 14]
and
which satisfy
The functions \(K(x)\) and \(E(x)\) have the following representation [23]:
and
where the hypergeometric function \(F ( a,b,c,x )\) is defined by [5]
with \(( a ) _{n}=\frac{\Gamma (a+n)}{\Gamma (a)} \) and the Euler gamma function \(\Gamma (x)\) is defined by the improper integral
The hypergeometric function \(F ( a,b,c,x ) \) has the differentiation formula [5]
and the transformation
Wallis’s ratio \(W_{n}\) is defined as [10, 12]
and satisfies the recurrence relation
In [31], Yang et al. show that
\(K(x)\) can be written using the notation \(W_{n}\) as follows:
The importance of elliptic integrals led to deduction of many of their inequalities. In [11], Carlson and Gustafson presented the inequality
In [16], Kühnau deduced the lower bound
which is an improvement of the left-hand side of inequality (11). In [4], Anderson et al. deduced the inequality
Alzer and Qiu [1] presented the inequality
with the best possible constants \(\mu = 3/4\) and \(\nu =1\), which improved the lower bound of (13). In [31], Yang et al. proved the inequality
In 2019, Yang and Tian [32] deduced the inequality
with the best possible constants \(\rho =\frac{\pi }{2 \ln 5}\) and \(\sigma =1\). Recently, Wang et al. [27] presented the inequality
For more details about inequalities, applications, and other related special functions to \(K(x)\) and \(E(x)\), we refer to [2, 3, 13, 15, 17–22, 24–26, 28–30] and the references therein.
Padé approximant [6–8] of order \((r,s)\) of a function \(f(x)\) is a rational function
where singularities of \(f(x)\) are only poles. There are many different ways to determine the other coefficients \(\alpha _{j}s\) for \(0 \leq j \leq r\) and \(\beta _{k}s\) for \(1 \leq k \leq s\). Among them is the matching between the first \(r+s+1\) coefficients in Maclaurin series \(f(x)=\sum_{k=0}^{\infty }c_{k} x^{k}\) and the first \(r+s+1\) coefficients of Padé approximant by the relation
Hence, we solve the following equations for \(\alpha _{i}s\) and \(\beta _{i}s\):
and we have
2 Main results
Theorem 1
The following inequality
holds for the best possible constant \(p=1\).
Proof
Consider the function
and then the function \(F_{p}(\sqrt{x})\) is strictly decreasing on \(x\in (0,1)\) if and only if
Using relation (7), we have
and hence
where
From (8), we have
where
Using (9), we obtain
and \(\mu _{0}=-1\), \(\mu _{1}=\frac{-1}{4}\), \(\mu _{2}=\frac{-17}{128}\), \(\mu _{3}=\frac{-43}{512}\), \(\mu _{4}=\frac{-953}{16{,}384}\), \(\mu _{5}=\frac{-2801}{65{,}536}\), \(\mu _{6}=\frac{-137{,}401}{4{,}194{,}304}\), \(\mu _{7}=\frac{-485{,}318}{16{,}777{,}216}\). Hence \(\mu _{n}<0\) for \(n\geqslant 0\), \(f_{1}(x)>0\) and therefore the function \(f(x)\) is increasing on \(x \in (0,1)\) with
which implies that \(p\leq 1\). Therefore, the function \(F_{p}(x)\) is strictly decreasing on \(x \in (0,1)\) if and only if \(p\leq 1\), and using the first limit in (3), we obtain inequality (18). □
Theorem 2
The following inequality
holds for the best possible constant \(q=\frac{5}{6}\).
Proof
Consider the function
and then the function \(H_{q}(\sqrt{x})\) is strictly increasing on \((0,1)\) if and only if
Then
where
From (8), we have
and
The sequence \(V_{n}<0\) for \(n=4,5,6,\ldots \) and
Then \(V_{n}<0\) for \(n\geqslant 0\), \(h_{1}(x)<0\) and therefore the function \(h(x)\) is decreasing with
which implies that \(q \geq \frac{5}{6}\). Therefore, the function \(H_{q}(x)\) is strictly increasing on \(x\in (0,1)\) if and only if \(q\geqslant \frac{5}{6}\), and using the limits in (3), we obtain inequality (19). □
Based on the Padé approximation method, we can conclude the following approximations.
Proposition 3
The Padé approximations of orders \((3, 4)\) and \((3,7)\) of the function
are the following rational functions:
and
Proposition 4
The Padé approximations of orders \((3, 7)\) and \((3,9)\) of the function
are the following rational functions:
and
Unfortunately, formulas (20),(21), (22), and (23) did not give bounds of the function \(K(x)\) for all x in the domain \((0,1)\). But formula (20) motivates us to establish the following inequalities.
Theorem 5
The following inequality
holds for \(0 < x<1\).
Proof
Consider the function
and hence
where
with
and
Now
and
Then \(w_{1} ( x ) \) is a convex function between the points \((0,-1)\) and \((1,0)\). Also,
where
and
where
Then \(w_{2} ( x ) \) is a convex function between the same two points \((0,-1)\) and \((1,0)\). Also,
and
Then \(w_{1} ( x ) > w_{2} ( x )\), \(t_{1}(x)>0\) and \(T(x)\) is decreasing on \(x \in (0,1)\). Hence, using the limits in (3), we obtain inequality (24). □
Theorem 6
The inequality
holds for \(x \in (0,1)\).
Proof
Consider the function
and hence
where
Using (5), we get
Now let
where
where
Then \(g_{2}(x)\) is deceasing with
Hence \(g_{2}(x)<0\), and we have
From inequalities (26) and (27), we get \(g_{1}(x)>0\) and the function \(G(x)\) is increasing. Hence, using the limits in (3), we obtain inequality (25) □
3 Remarks
Comparing our new bounds of the function \(K(x)\) with its previous ones presents the following remarks.
Remark 7
Our upper bound in (24) is better than our upper bound in (18) for \(x \in (0,1)\).
Remark 8
The upper bound in (24) is better than the upper bound in (11) for \(x \in (0,0.97)\).
Remark 9
The upper bound in (24) is better than the upper bound in (14) for \(x \in (0,1)\).
Remark 10
The upper bound in (24) is better than the upper bound in each of (15), (16), and (17) for \(x \in (0,0.98)\).
Remark 11
Our lower bounds in (19) and (25) are not contained in each other for \(x \in (0,1)\).
Remark 12
Our lower bound in (25) is better than the lower bound in (12) for \(x \in (0,9)\).
Remark 13
The lower bound in (25) is better than the lower bound in (14) for \(x \in (0,87)\).
Remark 14
The lower bound in (25) is better than the lower bound in (15) for \(x \in (0,94)\).
Remark 15
The lower bound in (25) is better than the lower bound in (16) for \(x \in (0,91)\).
Availability of data and materials
The data used to support the findings of this study are available from the corresponding author upon request.
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Mahmoud, M., Anis, M. Some Padé approximations and inequalities for the complete elliptic integrals of the first kind. J Inequal Appl 2021, 37 (2021). https://doi.org/10.1186/s13660-021-02568-0
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DOI: https://doi.org/10.1186/s13660-021-02568-0