Let $${\mathcal {K}}\left( r\right) $$ be the complete elliptic integrals of the first kind and $$\text{ arth }r$$ denote the inverse hyperbolic tangent function. We prove that the inequality $$\begin{aligned} \frac{2}{\pi }{\mathcal {K}}\left( r\right) >\left[ 1-\lambda +\lambda \left( \frac{\text{ arth }r}{r}\right) ^{q}\right] ^{1/q} \end{aligned}$$ holds for $$r\in \left( 0,1\right) $$ with the best constants $$\lambda =3/4$$ and $$q=1/10$$ . This improves some known results and gives a positive answer for a conjecture on the best upper bound for the Gaussian arithmetic–geometric mean in terms of logarithmic and arithmetic means.

中文翻译：

---

第一类完全椭圆积分的尖锐下界

令 $${\mathcal {K}}\left( r\right) $$ 是第一类完全椭圆积分，$$\text{ arth }r$$ 表示反双曲正切函数。我们证明不等式 $$\begin{aligned} \frac{2}{\pi }{\mathcal {K}}\left( r\right) >\left[ 1-\lambda +\lambda \left( \ frac{\text{ arth }r}{r}\right) ^{q}\right] ^{1/q} \end{aligned}$$ 保持 $$r\in \left( 0,1\right ) $$ 具有最佳常量 $$\lambda =3/4$$ 和 $$q=1/10$$ 。这改进了一些已知的结果，并对高斯算术-几何均值的最佳上界的猜想给出了肯定的答案，即对数和算术均值。