Digital Library of Mathematical Functions
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19 Elliptic IntegralsSymmetric Integrals

§19.24 Inequalities


§19.24(i) Complete Integrals

The condition yz 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 ln4zRF(0,y,z)+lny/z12π,
19.24.2 12z-1/2RG(0,y,z)14π,
19.24.3 (y3/2+z3/22)2/34πRG(0,y2,z2)(y2+z22)1/2,
y>0, z>0.

If y, z, and p are positive, then

19.24.4 2p(2yz+yp+zp)-1/243πRJ(0,y,z,p)(yzp2)-3/8.

Inequalities for RD(0,y,z) are included as the case p=z.

A series of successively sharper inequalities is obtained from the AGM process (§19.8(i)) with a0g0>0:

19.24.5 1an2πRF(0,a02,g02)1gn,


19.24.6 an+1 =(an+gn)/2,
gn+1 =angn.

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 RG(0,y,z) 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)=8RG(0,a2,b2).

For x>0, y>0, and xy, the complete cases of RF and RG satisfy

19.24.8 RF(x,y,0)RG(x,y,0) >18π2,
RF(x,y,0)+2RG(x,y,0) >π.

Also, with the notation of (19.24.6),

19.24.9 12g12RG(a02,g02,0)RF(a02,g02,0)12a12,

with equality iff a0=g0.

§19.24(ii) Incomplete Integrals

Inequalities for R-a(b;z) 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.


19.24.10 3x+y+zRF(x,y,z)1(xyz)1/6,
19.24.11 (5x+y+z+2p)3RJ(x,y,z,p)(xyzp2)-3/10,
19.24.12 13(x+y+z)RG(x,y,z)min(x+y+z3,x2+y2+z23xyz).

Inequalities for RC(x,y) and RD(x,y,z) are included as special cases (see (19.16.6) and (19.16.5)).

Other inequalities for RF(x,y,z) are given in Carlson (1970).

If a (0) is real, all components of b and z are positive, and the components of z are not all equal, then

19.24.13 Ra(b;z)R-a(b;z) >1,
Ra(b;z)+R-a(b;z) >2;

see Neuman (2003, (2.13)). Special cases with a=±12 are (19.24.8) (because of (19.16.20), (19.16.23)), and

19.24.14 RF(x,y,z)RG(x,y,z) >1,
RF(x,y,z)+RG(x,y,z) >2.

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 RC(x,12(y+z))RF(x,y,z)RC(x,yz),

with equality iff y=z.