1. ## Bessel function inequality

Can anybody help me to prove
$2x(K_1(x) - K_0(x)) - K_0(x) < 0$
where $x > 0$ and $K_i$ is the modified Bessel function of the second kind.

I solved it myself by noting that $K_1(x) = -K'_0(x)$, isolating $K'_0(x)/K_0(x)$, and applying Lemma 1 in Pal’tsev (1999) "Two-Sided Bounds Uniform in the Real Argument and the Index for Modified Bessel Functions."