Can anybody help me to prove
$\displaystyle 2x(K_1(x) - K_0(x)) - K_0(x) < 0$
where $\displaystyle x > 0$ and $\displaystyle K_i$ is the modified Bessel function of the second kind.
Thanks in advance!
I solved it myself by noting that $\displaystyle K_1(x) = -K'_0(x)$, isolating $\displaystyle K'_0(x)/K_0(x)$, and applying Lemma 1 in Pal’tsev (1999) "Two-Sided Bounds Uniform in the Real Argument and the Index for Modified Bessel Functions."