§10.14 Inequalities; Monotonicity

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Keywords:: Bessel functions, inequalities, monotonicity
Notes:: See Watson (1944, pp. 49, 258–259, 268–270, 406) and Olver (1997b, pp. 59, 426).
Referenced by:: Erratum (V1.0.1) for References
Permalink:: http://dlmf.nist.gov/10.14
Addition (effective with 1.0.1):: An additional reference was inserted in the last sentence of this section.
See also:: Annotations for Ch.10

10.14.1		$\displaystyle\|J_{\nu}\left(x\right)\|$	$\displaystyle\leq 1,$
	$\nu\geq 0,x\in\mathbb{R}$ ,
		$\displaystyle\|J_{\nu}\left(x\right)\|$	$\displaystyle\leq 2^{-\frac{1}{2}},$
	$\nu\geq 1,x\in\mathbb{R}$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\in$ : element of, $\mathbb{R}$ : real line, $x$ : real variable and $\nu$ : complex parameter A&S Ref: 9.1.60 Permalink: http://dlmf.nist.gov/10.14.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.14 and Ch.10

10.14.2		$0<J_{\nu}\left(\nu\right)<\frac{2^{\frac{1}{3}}}{3^{\frac{2}{3}}\Gamma\left(% \tfrac{2}{3}\right)\nu^{\frac{1}{3}}},$
$\nu>0$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function and $\nu$ : complex parameter A&S Ref: 9.1.61 Permalink: http://dlmf.nist.gov/10.14.E2 Encodings: TeX, pMML, png See also: Annotations for §10.14 and Ch.10

For monotonicity properties of $J_{\nu}\left(\nu\right)$ and $J_{\nu}'\left(\nu\right)$ see Lorch (1992).

10.14.3	$\displaystyle\|J_{n}\left(z\right)\|$	$\displaystyle\leq e^{\|\Im z\|},$
$n\in\mathbb{Z}$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\Im$ : imaginary part, $\mathbb{Z}$ : set of all integers, $n$ : integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/10.14.E3 Encodings: TeX, pMML, png See also: Annotations for §10.14 and Ch.10
10.14.4	$\displaystyle\|J_{\nu}\left(z\right)\|$	$\displaystyle\leq\frac{\|\tfrac{1}{2}z\|^{\nu}e^{\|\Im z\|}}{\Gamma\left(\nu+1% \right)},$
$\nu\geq-\frac{1}{2}$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{e}$ : base of natural logarithm, $\Im$ : imaginary part, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.62 (corrected) Permalink: http://dlmf.nist.gov/10.14.E4 Encodings: TeX, pMML, png See also: Annotations for §10.14 and Ch.10

10.14.5		$\|J_{\nu}\left(\nu x\right)\|\leq\frac{x^{\nu}\exp\left(\nu(1-x^{2})^{\frac{1}{2% }}\right)}{\left(1+(1-x^{2})^{\frac{1}{2}}\right)^{\nu}},$
$\nu\geq 0,0<x\leq 1$ ;
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\exp\NVar{z}$ : exponential function, $x$ : real variable and $\nu$ : complex parameter Permalink: http://dlmf.nist.gov/10.14.E5 Encodings: TeX, pMML, png See also: Annotations for §10.14 and Ch.10

see Siegel (1953).

10.14.6		$\|J_{\nu}'\left(\nu x\right)\|\leq\frac{(1+x^{2})^{\frac{1}{4}}}{x(2\pi\nu)^{% \frac{1}{2}}}\frac{x^{\nu}\exp\left(\nu(1-x^{2})^{\frac{1}{2}}\right)}{\left(1% +(1-x^{2})^{\frac{1}{2}}\right)^{\nu}},$
$\nu>0,0<x\leq 1$ ;
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $x$ : real variable and $\nu$ : complex parameter Permalink: http://dlmf.nist.gov/10.14.E6 Encodings: TeX, pMML, png See also: Annotations for §10.14 and Ch.10

see Watson (1944, p. 255). For a related bound for $Y_{\nu}\left(\nu x\right)$ see Siegel and Sleator (1954).

10.14.7		$1\leq\frac{J_{\nu}\left(\nu x\right)}{x^{\nu}J_{\nu}\left(\nu\right)}\leq e^{% \nu(1-x)},$
$\nu\geq 0,0<x\leq 1$ ;
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\mathrm{e}$ : base of natural logarithm, $x$ : real variable and $\nu$ : complex parameter Permalink: http://dlmf.nist.gov/10.14.E7 Encodings: TeX, pMML, png See also: Annotations for §10.14 and Ch.10

see Paris (1984). For similar bounds for $\mathscr{C}_{\nu}\left(x\right)$ (§10.2(ii)) see Laforgia (1986).

Kapteyn’s Inequality

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10.14.8		$\|J_{n}\left(nz\right)\|\leq\frac{\left\|z^{n}\exp\left(n(1-z^{2})^{\frac{1}{2}}% \right)\right\|}{\left\|1+(1-z^{2})^{\frac{1}{2}}\right\|^{n}},$
$n=0,1,2,\dots$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\exp\NVar{z}$ : exponential function, $n$ : integer and $z$ : complex variable A&S Ref: 9.1.63 Permalink: http://dlmf.nist.gov/10.14.E8 Encodings: TeX, pMML, png See also: Annotations for §10.14, §10.14 and Ch.10

where $(1-z^{2})^{\frac{1}{2}}$ has its principal value.

10.14.9		$\|J_{n}\left(nz\right)\|\leq 1,$
$n=0,1,2,\dots,z\in\mathbf{K}$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\in$ : element of, $n$ : integer, $z$ : complex variable and $\mathbf{K}$ : domain Permalink: http://dlmf.nist.gov/10.14.E9 Encodings: TeX, pMML, png See also: Annotations for §10.14, §10.14 and Ch.10

where $\mathbf{K}$ is defined in §10.20(ii).

For inequalities for the function $\Gamma\left(\nu+1\right)(2/x)^{\nu}J_{\nu}\left(x\right)$ with $\nu>-\tfrac{1}{2}$ see Neuman (2004).

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