# §10.14 Inequalities; Monotonicity

 10.14.1 $\displaystyle|\mathop{J_{\nu}\/}\nolimits\!\left(x\right)|$ $\displaystyle\leq 1,$ $\nu\geq 0,x\in\mathbb{R}$, $\displaystyle|\mathop{J_{\nu}\/}\nolimits\!\left(x\right)|$ $\displaystyle\leq 2^{-\frac{1}{2}},$ $\nu\geq 1,x\in\mathbb{R}$. Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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
 10.14.2 $0<\mathop{J_{\nu}\/}\nolimits\!\left(\nu\right)<\frac{2^{\frac{1}{3}}}{3^{% \frac{2}{3}}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{2}{3}\right)\nu^{\frac{1}% {3}}},$ $\nu>0$. Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: 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

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

 10.14.3 $\displaystyle|\mathop{J_{n}\/}\nolimits\!\left(z\right)|$ $\displaystyle\leq e^{|\Im{z}|},$ $n\in\mathbb{Z}$. 10.14.4 $\displaystyle|\mathop{J_{\nu}\/}\nolimits\!\left(z\right)|$ $\displaystyle\leq\frac{|\tfrac{1}{2}z|^{\nu}e^{|\Im{z}|}}{\mathop{\Gamma\/}% \nolimits\!\left(\nu+1\right)},$ $\nu\geq-\frac{1}{2}$.
 10.14.5 $|\mathop{J_{\nu}\/}\nolimits\!\left(\nu x\right)|\leq\frac{x^{\nu}\mathop{\exp% \/}\nolimits\left(\nu(1-x^{2})^{\frac{1}{2}}\right)}{\left(1+(1-x^{2})^{\frac{% 1}{2}}\right)^{\nu}},$ $\nu\geq 0,0;

see Siegel (1953).

 10.14.6 $|\mathop{J_{\nu}\/}\nolimits'\!\left(\nu x\right)|\leq\frac{(1+x^{2})^{\frac{1% }{4}}}{x(2\pi\nu)^{\frac{1}{2}}}\frac{x^{\nu}\mathop{\exp\/}\nolimits\left(\nu% (1-x^{2})^{\frac{1}{2}}\right)}{\left(1+(1-x^{2})^{\frac{1}{2}}\right)^{\nu}},$ $\nu>0,0;

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

 10.14.7 $1\leq\frac{\mathop{J_{\nu}\/}\nolimits\!\left(\nu x\right)}{x^{\nu}\mathop{J_{% \nu}\/}\nolimits\!\left(\nu\right)}\leq e^{\nu(1-x)},$ $\nu\geq 0,0;

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

## Kapteyn’s Inequality

 10.14.8 $|\mathop{J_{n}\/}\nolimits\!\left(nz\right)|\leq\frac{\left|z^{n}\mathop{\exp% \/}\nolimits\!\left(n(1-z^{2})^{\frac{1}{2}}\right)\right|}{\left|1+(1-z^{2})^% {\frac{1}{2}}\right|^{n}},$ $n=0,1,2,\dots$, Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the first kind, $\mathop{\exp\/}\nolimits\NVar{z}$: 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

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

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

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

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

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