§10.14 Inequalities; Monotonicity

 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 10
 10.14.2 $0 $\nu>0$. ⓘ 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 10

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

 10.14.3 $\displaystyle|J_{n}\left(z\right)|$ $\displaystyle\leq e^{|\Im z|},$ $n\in\mathbb{Z}$. 10.14.4 $\displaystyle|J_{\nu}\left(z\right)|$ $\displaystyle\leq\frac{|\tfrac{1}{2}z|^{\nu}e^{|\Im z|}}{\Gamma\left(\nu+1% \right)},$ $\nu\geq-\frac{1}{2}$.
 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;

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;

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;

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

Kapteyn’s Inequality

 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 10

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

 10.14.9 $|J_{n}\left(nz\right)|\leq 1,$ $n=0,1,2,\dots,z\in\mathbf{K}$,

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).