Relaxed super self-duality and effective action

A closed-form expression is obtained for a holomorphic sector of the two-loop Euler-Heisenberg type effective action for N = 2 supersymmetric QED derived in hep-th/0308136. In the framework of the background-field method, this sector is singled out by computing the effective action for a background N = 2 vector multiplet satisfying a relaxed super self-duality condition. The approach advocated in this letter can be applied, in particular, to the study of the N = 4 super Yang-Mills theory on its Coulomb branch.