[Paper Review] Asymptotical Bounds for Complete Elliptic Integrals of the Second Kind

Miao-Kun Wang, Yu‐Ming Chu|arXiv (Cornell University)|Sep 1, 2012

Analytic Number Theory Research3 citations

TL;DR

This paper establishes sharp asymptotically precise upper and lower bounds for the complete elliptic integral of the second kind, $\mathcal{E}(r)$, improving upon known inequalities including Vuorinen's conjecture and those by Alzer and Qiu. Using monotonicity analysis and integral identities, the authors derive double inequalities involving parameterized means that converge precisely to $\mathcal{E}(r)$ as $r \to 1^-$, with optimal parameters derived from asymptotic behavior and convexity arguments.

ABSTRACT

In this paper, we establish several asymptotical bounds for the complete elliptic integrals of the second kind $\mathcal{E}(r)$, and improve the well-known conjecture $\mathcal{E}(r)>π[(1+(1-r^2)^{3/4})/2]^{2/3}/2$ for all $r\in(0,1)$ proposed by M. Vuorinen.

Motivation & Objective

• To refine and improve existing asymptotic bounds for the complete elliptic integral of the second kind $\mathcal{E}(r)$, particularly those proposed by M. Vuorinen and others.
• To establish tighter upper and lower bounds that are asymptotically sharp as $r \to 1^-$, ensuring improved accuracy near the singularity point.
• To provide rigorous proofs for the optimality of the bounds by determining exact parameter thresholds using monotonicity and convexity properties.
• To compare the new bounds with classical inequalities (e.g., (1.1), (1.2), (1.3)) and demonstrate their superiority in the limit $r \to 1^-$.
• To extend the framework of mean-based approximations for $\mathcal{E}(r)$ using parameterized symmetric means and power-weighted expressions.

Proposed method

• Derives new double inequalities for $\mathcal{E}(r)$ using parameterized symmetric means of the form $\sqrt{a + (1-a)r'^2} + \sqrt{(1-a) + a r'^2}$, scaled by $\pi/4$.
• Applies the monotonicity criterion from Lemma 2.1 to analyze ratios of derivatives, enabling proof of strict monotonicity of key functions involving $\mathcal{E}(r)$ and $\mathcal{K}(r)$.
• Uses known integral identities and asymptotic expansions (e.g., $\lim_{r\to1} r'^\alpha \mathcal{K}(r) = 0$ for $\alpha > 0$) to determine boundary behavior.
• Introduces a generalized power-mean structure with parameter $p$, leading to bounds of the form $2^{p-2}\pi(1+r')^{1-2p} \left[ \cdots \right]^p$, which are optimized via critical parameter thresholds.
• Employs algebraic transformations and equivalence chains to compare bounds, reducing inequalities to polynomial sign analysis.
• Applies the two-point Gauss-Chebyshev quadrature remainder formula (from [11]) to validate earlier conjectured bounds, particularly (1.3).

Experimental results

Research questions

• RQ1 Can the conjectured lower bound $\mathcal{E}(r) > \frac{\pi}{2}\left(\frac{1 + r'^{3/2}}{2}\right)^{2/3}$ be improved with asymptotically sharper bounds?
• RQ2 What are the optimal parameters $\alpha, \beta$ such that the double inequality $\frac{\pi}{4}\left(\sqrt{\alpha + (1-\alpha)r'^2} + \sqrt{(1-\alpha) + \alpha r'^2}\right) < \mathcal{E}(r) < \cdots$ holds for all $r \in (0,1)$?
• RQ3 How do the new bounds compare asymptotically to known bounds (e.g., (1.1), (1.2), (1.3)) as $r \to 1^-$?
• RQ4 Can a unified framework of power-mean-type bounds be constructed that improves upon both upper and lower bounds simultaneously?
• RQ5 What is the exact threshold for the parameter $t$ in the generalized mean form such that the resulting bound is optimal and tight in the limit $r \to 1^-$?

Key findings

• The lower bound in Theorem 1.1 with $\beta = \frac{1}{2} - \frac{2\sqrt{2(\pi^2 - 8)}}{\pi^2}$ satisfies $\lim_{r \to 1} \text{bound} = 1$, matching $\mathcal{E}(1^-) = 1$, thus being asymptotically sharp.
• The upper bound in Corollary 3.1 with $\mu = \frac{1}{2} + \frac{1}{2}\sqrt{(4/\pi)^2 - 1}$ satisfies $\lim_{r \to 1} \text{bound} = 1$, achieving exact asymptotic convergence.
• The upper bound in Corollary 3.1 is strictly better than the bound in (1.2) for all $r \in (0,1)$, as shown by the inequality $ (1+x^2) > \left[\mu + (1-\mu)x\right]^2 + \left[(1-\mu) + \mu x\right]^2 $ for $x = r'$.
• The lower bound in Corollary 3.1 with $\lambda = \frac{1}{2} + \frac{\sqrt{2}}{8}$ strictly exceeds the Vuorinen conjectured bound $\frac{\pi}{2}\left(\frac{1 + r'^{3/2}}{2}\right)^{2/3}$ for all $r \in (0,1)$, as verified via polynomial sign analysis.
• The upper bound in Theorem 1.1 with $\alpha = \frac{1}{2} - \frac{\sqrt{2}}{4}$ coincides exactly with the Alzer-Qiu bound (1.3), confirming its optimality and providing a new proof via monotonicity.
• For $r \in (1 - \delta_1, 1)$ with $\delta_1 > 0$, the upper bound in Corollary 3.1 is tighter than (1.3), and for $r \in (1 - \delta_2, 1)$ with $\delta_2 > 0$, the lower bound in Theorem 1.1 is tighter than (1.1), demonstrating local superiority near $r=1$.