# §19.9 Inequalities

## §19.9(i) Complete Integrals

Throughout this subsection $0, except in (19.9.4).

 19.9.1 $\displaystyle\ln 4$ $\displaystyle\leq K\left(k\right)+\ln k^{\prime}\leq\pi/2,$ $\displaystyle 1$ $\displaystyle\leq E\left(k\right)\leq\pi/2.$ $\displaystyle 1$ $\displaystyle\leq(2/\pi)\sqrt{1-\alpha^{2}}\Pi\left(\alpha^{2},k\right)\leq 1/% k^{\prime},$ $\alpha^{2}<1$.
 19.9.2 $1+\frac{{k^{\prime}}^{2}}{8}<\frac{K\left(k\right)}{\ln\left(4/k^{\prime}% \right)}<1+\frac{{k^{\prime}}^{2}}{4},$ ⓘ Symbols: $K\left(\NVar{k}\right)$: Legendre’s complete elliptic integral of the first kind, $\ln\NVar{z}$: principal branch of logarithm function, $k$: real or complex modulus and $k^{\prime}$: complementary modulus Referenced by: §19.9(i), §19.9(i) Permalink: http://dlmf.nist.gov/19.9.E2 Encodings: TeX, pMML, png See also: Annotations for §19.9(i), §19.9 and Ch.19
 19.9.3 $9+\frac{k^{2}{k^{\prime}}^{2}}{8}<\frac{(8+k^{2})K\left(k\right)}{\ln\left(4/k% ^{\prime}\right)}<9.096.$ ⓘ Symbols: $K\left(\NVar{k}\right)$: Legendre’s complete elliptic integral of the first kind, $\ln\NVar{z}$: principal branch of logarithm function, $k$: real or complex modulus and $k^{\prime}$: complementary modulus Referenced by: §19.9(i), §19.9(i) Permalink: http://dlmf.nist.gov/19.9.E3 Encodings: TeX, pMML, png See also: Annotations for §19.9(i), §19.9 and Ch.19

The left-hand inequalities in (19.9.2) and (19.9.3) are equivalent, but the right-hand inequality of (19.9.3) is sharper than that of (19.9.2) when $0.

 19.9.4 $\left(\frac{1+{k^{\prime}}^{3/2}}{2}\right)^{2/3}\leq\frac{2}{\pi}E\left(k% \right)\leq\left(\frac{1+{k^{\prime}}^{2}}{2}\right)^{1/2}$

for $0\leq k\leq 1$. The lower bound in (19.9.4) is sharper than $2/\pi$ when $0\leq k^{2}\leq 0.9960$.

 19.9.5 $\ln\frac{(1+\sqrt{k^{\prime}})^{2}}{k}<\frac{\pi{K^{\prime}}\left(k\right)}{2K% \left(k\right)}<\ln\frac{2(1+k^{\prime})}{k}.$

For a sharper, but more complicated, version of (19.9.5) see Anderson et al. (1990).

Other inequalities are:

 19.9.6 $(1-\tfrac{3}{4}k^{2})^{-1/2}<\frac{4}{\pi k^{2}}(K\left(k\right)-E\left(k% \right))<(k^{\prime})^{-3/4},$
 19.9.7 $(1-\tfrac{1}{4}k^{2})^{-1/2}<\frac{4}{\pi k^{2}}(E\left(k\right)-{k^{\prime}}^% {2}K\left(k\right))<\min((k^{\prime})^{-1/4},4/\pi),$
 19.9.8 $k^{\prime}<\frac{E\left(k\right)}{K\left(k\right)}<\left(\frac{1+k^{\prime}}{2% }\right)^{2}.$

Further inequalities for $K\left(k\right)$ and $E\left(k\right)$ can be found in Alzer and Qiu (2004), Anderson et al. (1992a, b, 1997), and Qiu and Vamanamurthy (1996).

The perimeter $L(a,b)$ of an ellipse with semiaxes $a,b$ is given by

 19.9.9 $L(a,b)=4aE\left(k\right),$ $k^{2}=1-(b^{2}/a^{2})$, $a>b$. ⓘ Symbols: $E\left(\NVar{k}\right)$: Legendre’s complete elliptic integral of the second kind, $k$: real or complex modulus and $L(a,b)$: perimeter Referenced by: §19.9(i) Permalink: http://dlmf.nist.gov/19.9.E9 Encodings: TeX, pMML, png See also: Annotations for §19.9(i), §19.9 and Ch.19

Almkvist and Berndt (1988) list thirteen approximations to $L(a,b)$ that have been proposed by various authors. The earliest is due to Kepler and the most accurate to Ramanujan. Ramanujan’s approximation and its leading error term yield the following approximation to $L(a,b)/(\pi(a+b))$:

 19.9.10 $1+\frac{3\lambda^{2}}{10+\sqrt{4-3\lambda^{2}}}+\frac{3\lambda^{10}}{2^{17}},$ $\lambda=\dfrac{a-b}{a+b}$. ⓘ Permalink: http://dlmf.nist.gov/19.9.E10 Encodings: TeX, pMML, png See also: Annotations for §19.9(i), §19.9 and Ch.19

Even for the extremely eccentric ellipse with $a=99$ and $b=1$, this is correct within 0.023%. Barnard et al. (2000) shows that nine of the thirteen approximations, including Ramanujan’s, are from below and four are from above. See also Barnard et al. (2001).

## §19.9(ii) Incomplete Integrals

Throughout this subsection we assume that $0, $0\leq\phi\leq\pi/2$, and $\Delta=\sqrt{1-k^{2}{\sin}^{2}\phi}>0$.

Simple inequalities for incomplete integrals follow directly from the defining integrals (§19.2(ii)) together with (19.6.12):

 19.9.11 $\phi\leq F\left(\phi,k\right)\leq\min(\phi/\Delta,{\operatorname{gd}^{-1}}% \left(\phi\right)),$

where ${\operatorname{gd}^{-1}}\left(\phi\right)$ is given by (4.23.41) and (4.23.42). Also,

 19.9.12 $\max(\sin\phi,\phi\Delta)\leq E\left(\phi,k\right)\leq\phi,$
 19.9.13 $\Pi\left(\phi,\alpha^{2},0\right)\leq\Pi\left(\phi,\alpha^{2},k\right)\leq\min% (\Pi\left(\phi,\alpha^{2},0\right)/\Delta,\Pi\left(\phi,\alpha^{2},1\right)).$

Sharper inequalities for $F\left(\phi,k\right)$ are:

 19.9.14 $\frac{3}{1+\Delta+\cos\phi}<\frac{F\left(\phi,k\right)}{\sin\phi}<\frac{1}{(% \Delta\cos\phi)^{1/3}},$
 19.9.15 $1
 19.9.16 $F\left(\phi,k\right)=\frac{2}{\pi}K\left(k^{\prime}\right)\ln\left(\frac{4}{% \Delta+\cos\phi}\right)-\theta\Delta^{2},$ $(\sin\phi)/8<\theta<(\ln 2)/(k^{2}\sin\phi)$.

(19.9.15) is useful when $k^{2}$ and ${\sin}^{2}\phi$ are both close to $1$, since the bounds are then nearly equal; otherwise (19.9.14) is preferable.

Inequalities for both $F\left(\phi,k\right)$ and $E\left(\phi,k\right)$ involving inverse circular or inverse hyperbolic functions are given in Carlson (1961b, §4). For example,

 19.9.17 $L\leq F\left(\phi,k\right)\leq\sqrt{UL}\leq\tfrac{1}{2}(U+L)\leq U,$

where

 19.9.18 $\displaystyle L$ $\displaystyle=(1/\sigma)\operatorname{arctanh}\left(\sigma\sin\phi\right),$ $\sigma=\sqrt{(1+k^{2})/2}$, $\displaystyle U$ $\displaystyle=\tfrac{1}{2}\operatorname{arctanh}\left(\sin\phi\right)+\tfrac{1% }{2}k^{-1}\operatorname{arctanh}\left(k\sin\phi\right).$

Other inequalities for $F\left(\phi,k\right)$ can be obtained from inequalities for $R_{F}\left(x,y,z\right)$ given in Carlson (1966, (2.15)) and Carlson (1970) via (19.25.5).