Digital Library of Mathematical Functions
About the Project
NIST
10 Bessel FunctionsBessel and Hankel Functions

§10.14 Inequalities; Monotonicity

10.14.1 |Jν(x)| 1,
ν0,x,
|Jν(x)| 2-12,
ν1,x.
10.14.2 0<Jν(ν)<213323Γ(23)ν13,
ν>0.

For monotonicity properties of Jν(ν) and Jν(ν) see Lorch (1992).

10.14.3 |Jn(z)| |z|,
n.
10.14.4 |Jν(z)| |12z|ν|z|Γ(ν+1),
ν-12.
10.14.5 |Jν(νx)|xνexp(ν(1-x2)12)(1+(1-x2)12)ν,
ν0,0<x1;

see Siegel (1953).

10.14.6 |Jν(νx)|(1+x2)14x(2πν)12xνexp(ν(1-x2)12)(1+(1-x2)12)ν,
ν>0,0<x1;

see Watson (1944, p. 255). For a related bound for Yν(νx) see Siegel and Sleator (1954).

10.14.7 1Jν(νx)xνJν(ν)ν(1-x),
ν0,0<x1;

see Paris (1984). For similar bounds for 𝒞ν(x)10.2(ii)) see Laforgia (1986).

Kapteyn’s Inequality

10.14.8 |Jn(nz)||znexp(n(1-z2)12)||1+(1-z2)12|n,
n=0,1,2,,

where (1-z2)12 has its principal value.

10.14.9 |Jn(nz)|1,
n=0,1,2,,zK,

where K is defined in §10.20(ii).

For inequalities for the function Γ(ν+1)(2/x)νJν(x) with ν>-12 see Neuman (2004).

For further monotonicity properties see Landau (1999, 2000), and Muldoon and Spigler (1984).