  A closed-form expression is obtained for a holomorphic sector of the two-loop
low-energy effective action for the N = 4 super Yang-Mills theory on its
Coulomb branch where the gauge group SU(N) is spontaneously broken to SU(N-1) x
U(1) and the dynamics is described by a single Abelian N = 2 vector multiplet.
In the framework of the background-field method, this holomorphic sector is
singled out by computing the effective action for a background N = 2 vector
multiplet satisfying a relaxed super self-duality condition. At the two-loop
level, the N = 4 SYM effective action is shown to possess no F^4 term (with F
the U(1) field strength), in accordance with the Dine-Seiberg
non-renormalization conjecture hep-th/9705057 and its generalized form given in
hep-th/0310025. An unexpected outcome of our calculation is that no (manifestly
supersymmetric invariant generating) F^6 quantum correction occurs at two
loops. This is in conflict with previous results.
