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On approximating the arithmetic-geometric mean and complete elliptic integral of the first kind. (English) Zbl 1387.33029
Summary: In the article, we prove that the double inequalities \[ \begin{gathered} \frac{1 +(6 p - 7) r^\prime}{p +(5 p - 6) r^\prime} \frac{\pi \tanh^{- 1}(r)}{2 r} < \mathcal{K}(r) < \frac{1 +(6 q - 7) r^\prime}{q +(5 q - 6) r^\prime} \frac{\pi \tanh^{- 1}(r)}{2 r},\\ \frac{q A(1, r) +(5 q - 6) G(1, r)}{A(1, r) +(6 q - 7) G(1, r)} L(1, r) < A G M(1, r) < \frac{p A(1, r) +(5 p - 6) G(1, r)}{A(1, r) +(6 p - 7) G(1, r)} L(1, r)\end{gathered} \] hold for all \(r \in(0, 1)\) if and only if \(p \geq \pi / 2 = 1.570796 \cdots\) and \(q \leq 89 / 69 = 1.289855 \cdots\), where \(\mathcal{K}(r) = \int_0^{\pi / 2}(1 - r^2 \sin^2 t)^{- 1 / 2} d t\) is the complete elliptic integral of the first kind, \(\tanh^{- 1}(r) = \log [(1 + r) /(1 - r)] / 2\) is the inverse hyperbolic tangent function, \(r^\prime = \sqrt{1 - r^2}\), and \(A(1, r) = (1 + r) / 2\), \(G(1, r) = \sqrt{r}\), \(L(1, r) = (r - 1) / \log r\) and \(A G M(1, r)\) are the arithmetic, geometric, logarithmic and Gaussian arithmetic-geometric means of 1 and \(r\), respectively.

MSC:
33E05 Elliptic functions and integrals
26D15 Inequalities for sums, series and integrals
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