AI 中文总结
本文针对修正贝塞尔函数的乘积及比值，除证明了一个此前猜想的下界外，还通过渐近优化单参数不等式得到高精度最优界，可在宽参数范围提供精确估计。
AI 中文摘要
本文给出了修正贝塞尔函数的乘积及比值的新的精确界。乘积的大部分界是由已建立的连续阶比值的界直接推导而来，除了下界 $I_\ u(x)K_\ u(x) > \ rac{1}{2}(x^2 + \ u^2 + 1/5)^{-1/2}$，该下界此前被猜想适用于 $x > 0$ 且 $\ u > -1$，本文利用德拜型渐近法对大 $\ u$ 证明了该下界。此外，通过渐近优化某些单参数不等式，得到了比值（进而乘积）的非常精确的界。这些优化后的界精度极高：在固定 $\ u$ 时对小 $x$ 和大 $x$，以及固定 $x$ 时对大 $\ u$ 或固定 $z = x/\ u$ 的情况，都保持极高精度，因此能在广泛的参数范围内提供精确的上下估计。
英文摘要
New sharp bounds for the product and the ratios of modified Bessel functions are presented. Most bounds for the product are derived as direct consequences of previously established bounds for the ratios of consecutive orders, except for the lower bound $I_ν(x)K_ν(x) > \frac{1}{2}(x^2 + ν^2 + 1/5)^{-1/2}$, which had been conjectured for $x > 0$ and $ν> -1$ and we prove in the present paper, showing that the constant $1/5$ can not be lowered. Moreover, very sharp bounds are obtained for the ratios (and consequently for the product) by asymptotically optimizing certain uniparametric inequalities. These optimized bounds are remarkably accurate: they remain extremely sharp for both small and large $x$ with fixed $ν$, and for large $ν$ with fixed $x$ or fixed $z = x/ν$. As a consequence, they provide precise upper and lower estimates across a wide range of parameters.
CommentsA proof of the inequality $I_ν(x)K_ν(x) > \frac{1}{2}(x^2 + ν^2 + 1/5)^{-1/2}$ has been added (v2). Abstract corrected (v3)
