10 Bessel Functions Bessel and Hankel Functions 10.13 Other Differential Equations 10.15 Derivatives with Respect to Order

§10.14 Inequalities; Monotonicity

Notes:: See Watson (1944, pp. 49, 258–259, 268–270, 406) and Olver (1997b, pp. 59, 426).
Keywords:: Bessel functions, inequalities, monotonicity
Referenced by:: Other Changes
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.
Reported 2011-06-21

10.14.1

$|\mathop{J_{{\nu}}\/}\nolimits\!\left(x\right)|\leq 1,$ $\nu\geq 0,x\in\Real$ ,

$|\mathop{J_{{\nu}}\/}\nolimits\!\left(x\right)|\leq 2^{{-\frac{1}{2}}},$ $\nu\geq 1,x\in\Real$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\in$ : element of, $\Real$ : real line (excluding infinity), : 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

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:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function and $\nu$ : complex parameter
A&S Ref:: 9.1.61
Permalink:: http://dlmf.nist.gov/10.14.E2
Encodings:: TeX, pMML, png

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

10.14.3 $|\mathop{J_{{n}}\/}\nolimits\!\left(z\right)|\leq e^{{|\imagpart{z}|}},$ $n\in\Integer$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\in$ : element of, : base of exponential function, $\imagpart{}$ : imaginary part, $\Integer$ : set of all integers, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/10.14.E3
Encodings:: TeX, pMML, png

10.14.4 $|\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)|\leq\frac{|\tfrac{1}{2}z|^{\nu}% e^{{|\imagpart{z}|}}}{\mathop{\Gamma\/}\nolimits\!\left(\nu+1\right)},$ $\nu\geq-\frac{1}{2}$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : base of exponential function, $\imagpart{}$ : imaginary part, : 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

10.14.5 $|\mathop{J_{{\nu}}\/}\nolimits\!\left(\nu x\right)|\leq\frac{x^{\nu}\mathop{% \exp\/}\nolimits\left(\nu(1-x^{2})^{{\frac{1}{2}}}\right)}{\left(1+(1-x^{2})^{% {\frac{1}{2}}}\right)^{\nu}},$ $\nu\geq 0,0<x\leq 1$ ;

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\exp\/}\nolimits z$ : exponential function, : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.14.E5
Encodings:: TeX, pMML, png

see Siegel (1953).

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

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\exp\/}\nolimits z$ : exponential function, : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.14.E6
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : base of exponential function, : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.14.E7
Encodings:: TeX, pMML, png

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

¶ Kapteyn’s Inequality

Keywords:: Bessel functions, 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:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\exp\/}\nolimits z$ : exponential function, : integer and : complex variable
A&S Ref:: 9.1.63
Permalink:: http://dlmf.nist.gov/10.14.E8
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\in$ : element of, : integer, : complex variable and $\mathbf{K}$ : domain
Permalink:: http://dlmf.nist.gov/10.14.E9
Encodings:: TeX, pMML, png

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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 10.13 Other Differential Equations 10.15 Derivatives with Respect to Order