Digital Library of Mathematical Functions
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10 Bessel FunctionsModified Bessel Functions

§10.37 Inequalities; Monotonicity

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 0ν<μ and |phz|<π, then

10.37.1 |Kν(z)|<|Kμ(z)|.

See also Paltsev (1999), Petropoulou (2000) and Segura (2011).