We assume throughout that x,y, and z are nonnegative and at most one of them is 0. The last argument of RC, RD, and RJ is assumed to be positive (see (7) and (8)).
C1. RC(x,y) with ( x<< y. )
where 
1 with equalities if and only if x=0.
C2. RC(x,y) with y< < x. Two approximations of different accuracy are
where 
. The first approximation implies
F1. RF(x, y, z) with x, y < < z. 
. Then
where 
The upper bound implies
A sharper lower bound and a higher-order approximation are given by
By (12) this implies (since 4k2 < 4 - k'2 if k2 < 1)
where 
and 
.
F2. RF(x, y, z) with z < < x, y. Let a = (x+ y) / 2 and g = 
. Then
where 
, where AGM denotes Gauss's arithmetic-geometric mean [7, (6.10-6)(9.2-3),], and hence
with equalities if and only if x = y.
D1.RD(x, y, z) with x, y < < z. Let a = (x+ y ) /2 and g = . Then
where
.
D2. RD(x, y, z) with z < < x, y. Let a = (x+ y ) / 2 and g = . Then
where 
.
A higher-order approximation is
where 
.
An approximation of still higher order is
where we have used (11) and where 
,
D3.RD(x, y, z) with y, z < < x. Let a = (y+ z) / 2 and g =
. Then
where 
.
D4.RD(x, y, z) with x < < y, z. Let a = ( y+ z) / 2 and g = . Then
where 
.
J1.RJ(x, y, z, p) with x, y, z < < p. Let a = (x+ y+ z) / 3 and b = 
(xy+ xz+ yz)1/2. Then
where 
.
In the complete case a sharper result is
where 
(x+ y) / 2 ) with equalities if and only ifx= y.
J2.RJ(x, y, z, p) with p < < x, y, z. Let g = (xyz)1/3, 3h-1 = x-1+ y-1 + z-1, and 
. Note that g is the geometric mean and h is the harmonic mean, whence g 
h with equality if and only if x = y = z. Then
where 
A higher-order approximation is
where 
The second term in the approximation is independent of p but is otherwise as complicated as the function being approximated. The same is true of an even more accurate approximation [16, Thm. 11,] in which the error is of order p instead of pln p and the leading term involves RC.
J3. RJ(x,y,z,p) with x, y < < z, p. Let a = (x+ y) /2 and g = . Then
where 
J4.RJ(x,y,z,p with z, p< <x, y. Let a = (x+ y)/2, g = , b = 
, and d = (z+2p) /3. Then
where 
with equalities if and only if x = y. Since z << g, RC (z, g) can be estimated from (22). In the complete case (45) reduces to
with 
as before. A higher-order approximation is
where we have used (11) and where
.
J5.RJ(x, y, z, p with x< < y, z, p. Let a = (y+ z)/2 and g = . Then
where 
J6.RJ(x, y, z, p with y, z, p< < x. Let a = (y+ z)/2 and g = . Then
where 
In the complete case this reduces to
where 
G1.RG(x, y, z) with x, y< < z. Let a = (x+ y)/2 and g = . Then
where 
In the right-hand inequality it is assumed that 5a < z. A sharper result for the complete case is
where 
By (13) this follows from
G2.RG(x, y, z) with z< < x, y. Let a = (x+ y)/2 ) and g = . Then
where 
with equality if and only if x = y.
and 
