In the article, we prove that the double inequalities
$$ \frac{\sqrt{\pi}e^{-x}}{\sqrt{2(x+a)}}< K_{0}(x)< \frac{\sqrt{\pi }e^{-x}}{\sqrt{2(x+b)}},\qquad 1+ \frac{1}{2(x+a)}< \frac {K_{1}(x)}{K_{0}(x)}< 1+\frac{1}{2(x+b)} $$
hold for all
$x>0$
if and only if
$a\geq1/4$
and
$b=0$
if
$a, b\in[0, \infty)$
, where
$K_{\nu}(x)$
is the modified Bessel function of the second kind. As applications, we provide bounds for
$K_{n+1}(x)/K_{n}(x)$
with
$n\in\mathbb{N}$
and present the nece...
