Throughout this subsection , except in (19.9.4).
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 .
for . The lower bound in (19.9.4) is sharper than when .
For a sharper, but more complicated, version of (19.9.5) see Anderson et al. (1990).
Other inequalities are:
Further inequalities for and can be found in Alzer and Qiu (2004), Anderson et al. (1992a, b, 1997), and Qiu and Vamanamurthy (1996).
The perimeter of an ellipse with semiaxes is given by
Almkvist and Berndt (1988) list thirteen approximations to 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 :
Even for the extremely eccentric ellipse with and , 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).
Throughout this subsection we assume that , , and .
Simple inequalities for incomplete integrals follow directly from the defining integrals (§19.2(ii)) together with (19.6.12):
where is given by (4.23.41) and (4.23.42). Also,
Sharper inequalities for are:
(19.9.15) is useful when and are both close to , since the bounds are then nearly equal; otherwise (19.9.14) is preferable.
Inequalities for both and involving inverse circular or inverse hyperbolic functions are given in Carlson (1961b, §4). For example,