If ν (≥0) is fixed, then throughout the interval 0<x<∞,
Iν(x) is positive and increasing, and Kν(x) is
positive and decreasing.
If x (>0) is fixed, then throughout the interval 0<ν<∞,
Iν(x) is decreasing, and Kν(x) is increasing.
For sharper inequalities when the variables are real see Paris (1984)
and Laforgia (1991).
If 0≤ν<μ and |phz|<π, then
See also Pal′tsev (1999), Petropoulou (2000), Segura (2011) and Gaunt (2014).