and let denote an arbitrary small positive constant. Then as , with fixed,
where the branch of is determined by
We continue to use the notation of §10.17(i). Also, , , and for ,
Then as with fixed,
In the expansions (10.17.3) and (10.17.4) assume that and . Then the remainder associated with the sum does not exceed the first neglected term in absolute value and has the same sign provided that . Similarly for , provided that .
where denotes the variational operator (2.3.6), and the paths of variation are subject to the condition that changes monotonically. Bounds for are given by
As in §9.7(v) denote