论文标题
简单的界限具有最佳的精度，即改良的贝塞尔功能的比率
Simple bounds with best possible accuracy for ratios of modified Bessel functions
论文作者
论文摘要
表格$ b（α，β，γ，x）的最佳界限=（α+\ sqrt {β^2+γ^2+γ^2 x^2}）/x $用于修改的贝塞尔功能的比率：如果$α$，$β$和$γ$的$ b（yis a a a a a yimα，α，ac），x $ bessel函数的比例： $φ_ν（x）= i_ {ν-1}（x）/i_ν（x）$ as $ x \ rightarrow 0^ +$（分别$ x \ rightarrow +\ iffty $），函数$ b（α，β，β，γ，x，x）$ b（x，x）$ b（x（x）$ $ b（α，β，γ，x）$是任何正$ x $的$φ_ν（x）$的上部（分别较低），并且在$ x _*$时是最好的。比率$φ_ν（x）= k_ {ν+1}（x）/k_ν（x）$也是如此，但互换下限和上限（并且限制范围更大，$ν$）。在$ 0^+$和$+\ undty $的最大精度的界限中被恢复为$ x _*\ rightarrow 0^+$和$ x _*\ rightArrow+\ rightarrow+\ infty $，并且在这些情况下，系数具有简单的表达式。对于有限和正面$ x _*$的情况，我们提供了一个界限的界限家庭，这些家庭接近最佳界限并保留其汇合属性。
The best bounds of the form $B(α,β,γ,x)=(α+\sqrt{β^2+γ^2 x^2})/x$ for ratios of modified Bessel functions are characterized: if $α$, $β$ and $γ$ are chosen in such a way that $B(α,β,γ,x)$ is a sharp approximation for $Φ_ν(x)=I_{ν-1} (x)/I_ν(x)$ as $x\rightarrow 0^+$ (respectively $x\rightarrow +\infty$) and the graphs of the functions $B(α,β,γ,x)$ and $Φ_ν(x)$ are tangent at some $x=x_*>0$, then $B(α,β,γ,x)$ is an upper (respectively lower) bound for $Φ_ν(x)$ for any positive $x$, and it is the best possible at $x_*$. The same is true for the ratio $Φ_ν(x)=K_{ν+1} (x)/K_ν(x)$ but interchanging lower and upper bounds (and with a slightly more restricted range for $ν$). Bounds with maximal accuracy at $0^+$ and $+\infty$ are recovered in the limits $x_*\rightarrow 0^+$ and $x_*\rightarrow +\infty$, and for these cases the coefficients have simple expressions. For the case of finite and positive $x_*$ we provide uniparametric families of bounds which are close to the optimal bounds and retain their confluence properties.
