§19.24 Inequalities

Permalink:: http://dlmf.nist.gov/19.24

§19.24(i) Complete Integrals
§19.24(ii) Incomplete Integrals

§19.24(i) Complete Integrals

Notes:: The first three inequalities come from (19.9.1) and (19.9.4). For (19.24.4) see (19.16.22) and Carlson (1966, (2.15)). For (19.24.3) see (19.30.5). (19.24.8) is a special case of (19.24.13). For (19.24.9) see Neuman (2003, (4.2)).
Keywords:: inequalities, symmetric elliptic integrals
Referenced by:: Ch.19
Permalink:: http://dlmf.nist.gov/19.24.i

The condition $y\leq z$ for (19.24.1) and (19.24.2) serves only to identify $y$ as the smaller of the two nonzero variables of a symmetric function; it does not restrict validity.

19.24.1 $\mathop{\ln\/}\nolimits 4\leq\sqrt{z}\mathop{R_{F}\/}\nolimits\!\left(0,y,z\right)+\mathop{\ln\/}\nolimits\sqrt{y/z}\leq\tfrac{1}{2}\pi,$ $0<y\leq z$ ,

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind and $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function
Referenced by:: §19.24(i)
Permalink:: http://dlmf.nist.gov/19.24.E1
Encodings:: TeX, pMML, png

19.24.2 $\tfrac{1}{2}\leq z^{{-1/2}}\mathop{R_{G}\/}\nolimits\!\left(0,y,z\right)\leq\tfrac{1}{4}\pi,$ $0\leq y\leq z$ ,

Symbols:: $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind
Referenced by:: §19.24(i)
Permalink:: http://dlmf.nist.gov/19.24.E2
Encodings:: TeX, pMML, png

19.24.3 $\left(\frac{y^{{3/2}}+z^{{3/2}}}{2}\right)^{{2/3}}\leq\frac{4}{\pi}\mathop{R_{G}\/}\nolimits\!\left(0,y^{2},z^{2}\right)\leq\left(\frac{y^{2}+z^{2}}{2}\right)^{{1/2}},$ $y>0$ , $z>0$ .

Symbols:: $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind
Referenced by:: §19.24(i)
Permalink:: http://dlmf.nist.gov/19.24.E3
Encodings:: TeX, pMML, png

If $y$ , $z$ , and $p$ are positive, then

19.24.4 $\frac{2}{\sqrt{p}}(2yz+yp+zp)^{{-1/2}}\leq\frac{4}{3\pi}\mathop{R_{J}\/}\nolimits\!\left(0,y,z,p\right)\leq(yzp^{2})^{{-3/8}}.$

Symbols:: $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind
Referenced by:: §19.24(i)
Permalink:: http://dlmf.nist.gov/19.24.E4
Encodings:: TeX, pMML, png

Inequalities for $\mathop{R_{D}\/}\nolimits\!\left(0,y,z\right)$ are included as the case $p=z$ .

A series of successively sharper inequalities is obtained from the AGM process (§19.8(i)) with $a_{0}\geq g_{0}>0$ :

19.24.5 $\frac{1}{a_{n}}\leq\frac{2}{\pi}\mathop{R_{F}\/}\nolimits\!\left(0,a_{0}^{2},g_{0}^{2}\right)\leq\frac{1}{g_{n}},$ $n=0,1,2,\dots$ ,

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $n$ : nonnegative integer, $a_{n}$ : iterate and $g_{n}$ : iterate
Permalink:: http://dlmf.nist.gov/19.24.E5
Encodings:: TeX, pMML, png

where

19.24.6

$a_{{n+1}}=(a_{n}+g_{n})/2,$

$g_{{n+1}}=\sqrt{a_{n}g_{n}}.$

Symbols:: $n$ : nonnegative integer, $a_{n}$ : iterate and $g_{n}$ : iterate
Referenced by:: §19.24(i)
Permalink:: http://dlmf.nist.gov/19.24.E6
Encodings:: TeX, TeX, pMML, pMML, png, png

Other inequalities can be obtained by applying Carlson (1966, Theorems 2 and 3) to (19.16.20)–(19.16.23). Approximations and one-sided inequalities for $\mathop{R_{G}\/}\nolimits\!\left(0,y,z\right)$ follow from those given in §19.9(i) for the length $L(a,b)$ of an ellipse with semiaxes $a$ and $b$ , since

19.24.7 $L(a,b)=8\!\mathop{R_{G}\/}\nolimits\!\left(0,a^{2},b^{2}\right).$

Symbols:: $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind and $L(a,b)$ : length
Permalink:: http://dlmf.nist.gov/19.24.E7
Encodings:: TeX, pMML, png

For $x>0$ , $y>0$ , and $x\neq y$ , the complete cases of $\mathop{R_{F}\/}\nolimits$ and $\mathop{R_{G}\/}\nolimits$ satisfy

19.24.8

$\mathop{R_{F}\/}\nolimits\!\left(x,y,0\right)\mathop{R_{G}\/}\nolimits\!\left(x,y,0\right)>\tfrac{1}{8}\pi^{2},$

$\mathop{R_{F}\/}\nolimits\!\left(x,y,0\right)+2\!\mathop{R_{G}\/}\nolimits\!\left(x,y,0\right)>\pi.$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind and $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind
Referenced by:: ¶ ‣ §19.24(ii), §19.24(i)
Permalink:: http://dlmf.nist.gov/19.24.E8
Encodings:: TeX, TeX, pMML, pMML, png, png

Also, with the notation of (19.24.6),

19.24.9 $\frac{1}{2}\, g_{1}^{2}\leq\frac{\mathop{R_{G}\/}\nolimits\!\left(a_{0}^{2},g_{0}^{2},0\right)}{\mathop{R_{F}\/}\nolimits\!\left(a_{0}^{2},g_{0}^{2},0\right)}\leq\frac{1}{2}\, a_{1}^{2},$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind, $a_{n}$ : iterate and $g_{n}$ : iterate
Referenced by:: §19.24(i), §19.9(i)
Permalink:: http://dlmf.nist.gov/19.24.E9
Encodings:: TeX, pMML, png

with equality iff $a_{0}=g_{0}$ .

§19.24(ii) Incomplete Integrals

Keywords:: inequalities, symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.24.ii

Inequalities for $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ in Carlson (1966, Theorems 2 and 3) can be applied to (19.16.14)–(19.16.17). All variables are positive, and equality occurs iff all variables are equal.

¶ Examples

19.24.10 $\frac{3}{\sqrt{x}+\sqrt{y}+\sqrt{z}}\leq\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)\leq\frac{1}{(xyz)^{{1/6}}},$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.9(ii)
Permalink:: http://dlmf.nist.gov/19.24.E10
Encodings:: TeX, pMML, png

19.24.11 $\left(\frac{5}{\sqrt{x}+\sqrt{y}+\sqrt{z}+2\sqrt{p}}\right)^{3}\leq\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)\leq(xyzp^{2})^{{-3/10}},$

Symbols:: $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind
Permalink:: http://dlmf.nist.gov/19.24.E11
Encodings:: TeX, pMML, png

19.24.12 $\tfrac{1}{3}(\sqrt{x}+\sqrt{y}+\sqrt{z})\leq\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)\leq\min\left(\sqrt{\frac{x+y+z}{3}},\frac{x^{2}+y^{2}+z^{2}}{3\sqrt{xyz}}\right).$

Symbols:: $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind and $(a,b)$ : open interval
Permalink:: http://dlmf.nist.gov/19.24.E12
Encodings:: TeX, pMML, png

Inequalities for $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ and $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ are included as special cases (see (19.16.6) and (19.16.5)).

Other inequalities for $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ are given in Carlson (1970).

If $a$ ( $\neq 0$ ) is real, all components of $\mathbf{b}$ and $\mathbf{z}$ are positive, and the components of $z$ are not all equal, then

19.24.13

$\mathop{R_{{a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)>1,$

$\mathop{R_{{a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)+\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)>2;$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function
Referenced by:: §19.24(i)
Permalink:: http://dlmf.nist.gov/19.24.E13
Encodings:: TeX, TeX, pMML, pMML, png, png

see Neuman (2003, (2.13)). Special cases with $a=\pm\frac{1}{2}$ are (19.24.8) (because of (19.16.20), (19.16.23)), and

19.24.14

$\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)>1,$

$\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)+\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)>2.$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind and $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind
Permalink:: http://dlmf.nist.gov/19.24.E14
Encodings:: TeX, TeX, pMML, pMML, png, png

The same reference also gives upper and lower bounds for symmetric integrals in terms of their elementary degenerate cases. These bounds include a sharper but more complicated lower bound than that supplied in the next result:

19.24.15 $\mathop{R_{C}\/}\nolimits\!\left(x,\tfrac{1}{2}(y+z)\right)\leq\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)\leq\mathop{R_{C}\/}\nolimits\!\left(x,\sqrt{yz}\right),$ $x\geq 0$ ,

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind
Permalink:: http://dlmf.nist.gov/19.24.E15
Encodings:: TeX, pMML, png

with equality iff $y=z$ .

§19.24 Inequalities

Contents

§19.24(i) Complete Integrals

§19.24(ii) Incomplete Integrals

¶ Examples