§10.37 Inequalities; Monotonicity

Notes:: See Olver (1997b, pp. 251–252). For (10.37.1) see Everitt and Jones (1977).
Keywords:: inequalities, modified Bessel functions, monotonicity
Permalink:: http://dlmf.nist.gov/10.37

If $\nu$ $(\geq 0)$ is fixed, then throughout the interval $0<x<\infty$ , $\mathop{I_{{\nu}}\/}\nolimits\!\left(x\right)$ is positive and increasing, and $\mathop{K_{{\nu}}\/}\nolimits\!\left(x\right)$ is positive and decreasing.

If $x$ $(>0)$ is fixed, then throughout the interval $0<\nu<\infty$ , $\mathop{I_{{\nu}}\/}\nolimits\!\left(x\right)$ is decreasing, and $\mathop{K_{{\nu}}\/}\nolimits\!\left(x\right)$ is increasing.

For sharper inequalities when the variables are real see Paris (1984) and Laforgia (1991).

If $0\leq\nu<\mu$ and $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ , then

10.37.1 $|\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)|<|\mathop{K_{{\mu}}\/}\nolimits\!\left(z\right)|.$

Symbols:: $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.37
Permalink:: http://dlmf.nist.gov/10.37.E1
Encodings:: TeX, pMML, png

See also Pal ${}^{{\prime}}$ tsev (1999) and Petropoulou (2000).