10 Bessel FunctionsBessel and Hankel Functions10.13 Other Differential Equations10.15 Derivatives with Respect to Order

§10.14 Inequalities; Monotonicity

ⓘ

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		$|J_{\nu }\left(x\right)|$	$\le 1,$
	$\nu \ge 0,x\in ℝ$,
		$|J_{\nu }\left(x\right)|$	$\le 2^{-\frac{1}{2}},$
	$\nu \ge 1,x\in ℝ$.
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\in$: element of, $ℝ$: 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
$\nu >0$.
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: 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 $J_{\nu }^{\prime }\left(\nu \right)$ see Lorch (1992).

10.14.3		$|J_n\left(z\right)|$	$\le {\mathrm{e}}^{|\mathrm{\Im }z|},$
$n\in \mathbb{Z}$.
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: 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		$|J_{\nu }\left(z\right)|$	$\le {\displaystyle \frac{{|\frac{1}{2}z|}^{\nu }{\mathrm{e}}^{|\mathrm{\Im }z|}}{\mathrm{\Gamma }\left(\nu +1\right)}},$
$\nu \ge -\frac{1}{2}$.
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: 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		$$|J_{\nu }\left(\nu x\right)|\le \frac{x^{\nu }\exp \left(\nu {(1-x^2)}^{\frac{1}{2}}\right)}{{\left(1+{(1-x^2)}^{\frac{1}{2}}\right)}^{\nu }},$$
;
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\exp z$: 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		$$|J_{\nu }^{\prime }\left(\nu x\right)|\le \frac{{(1+x^2)}^{\frac{1}{4}}}{x{(2\pi \nu )}^{\frac{1}{2}}}\frac{x^{\nu }\exp \left(\nu {(1-x^2)}^{\frac{1}{2}}\right)}{{\left(1+{(1-x^2)}^{\frac{1}{2}}\right)}^{\nu }},$$
;
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: 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		$$1\le \frac{J_{\nu }\left(\nu x\right)}{x^{\nu }{J}_{\nu }\left(\nu \right)}\le {\mathrm{e}}^{\nu (1-x)},$$
;
ⓘ Symbols: $J_{\nu }\left(z\right)$: 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 ${𝒞}_{\nu }\left(x\right)$ (§10.2(ii)) see Laforgia (1986).

Kapteyn’s Inequality

ⓘ

Keywords:

Bessel functions, Kapteyn’s inequality

See also:

Annotations for §10.14 and Ch.10

10.14.8		$$|J_n\left(nz\right)|\le \frac{\left|z^n\exp \left(n{(1-z^2)}^{\frac{1}{2}}\right)\right|}{{\left|1+{(1-z^2)}^{\frac{1}{2}}\right|}^n},$$
$n=0,1,2,\mathrm{\dots }$,
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\exp z$: 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		$$|J_n\left(nz\right)|\le 1,$$
$n=0,1,2,\mathrm{\dots },z\in 𝐊$,
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\in$: element of, $n$: integer, $z$: complex variable and $𝐊$: 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 $𝐊$ is defined in §10.20(ii).

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

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

10.13 Other Differential Equations10.15 Derivatives with Respect to Order

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