Some Padé approximations and inequalities for the complete elliptic integrals of the first kind

In this paper, we present Padé approximations of some functions involving complete elliptic integrals of the first kind K(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K(x)$\end{document}, and motivated by these approximations we also present the following double inequality: 1−x21−x2+x462<2e2πK(x)−1(1+11−x2)<1−96100x21−96100x2+x464,x∈(0,1).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{1-x^{2}}{1-x^{2}+\frac{x^{4}}{62}}< \frac{2 e^{\frac{2}{\pi }K(x)-1}}{ (1+\frac{1}{\sqrt{1-x^{2}}} )}< \frac{1-\frac{96}{100}x^{2}}{1-\frac{96}{100}x^{2}+\frac{x^{4}}{64}},\quad x\in ( 0,1 ). $$\end{document} Our results have superiority over some new recent results.


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 = (a+n) (a) and the Euler gamma function (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] W n = (n + 1/2) ( 1 2 ) (n + 1) , n ∈ N, and satisfies the recurrence relation In [31], Yang et al. show that n-k (k + 1)(nk + 1) -6(2n + 1)W 2 n (n + 1)(n + 2) < 0, n ≥ 8.
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 μ = 3/4 and ν = 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 ρ = π 2 ln 5 and σ = 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][7][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 α j s for 0 ≤ j ≤ r and β k s for 1 ≤ k ≤ s. Among them is the matching between the first r + s + 1 coefficients in Maclaurin series f (x) = ∞ k=0 c k x k and the first r + s + 1 coefficients of Padé approximant by the relation Hence, we solve the following equations for α i s and β i s:

Theorem 1 The following inequality
holds for the best possible constant p = 1.

Theorem 2
The following inequality holds for the best possible constant q = 5 6 .
The sequence V n < 0 for n = 4, 5, 6, . . . and Then V n < 0 for n ≥ 0, h 1 (x) < 0 and therefore the function h(x) is decreasing with which implies that q ≥ 5 6 . Therefore, the function H q (x) is strictly increasing on x ∈ (0, 1) if and only if q ≥ 5 6 , and using the limits in (3), we obtain inequality (19).
Based on the Padé approximation method, we can conclude the following approximations. (3,4) and (3,7) of the function

Remarks
Comparing our new bounds of the function K(x) with its previous ones presents the following remarks.