Let R = Delta (0) \ U-n an be a Zalcman domain (or L-domain), where Delta (0) : 0 < /z/ < 1, Delta (n) : /z - c(n)/ less than or equal to r(n), c(n) SE arrow 0, Delta (n) subset of Delta (0) and Delta (n) boolean AND Delta (m) = phi (n not equal m). For an unlimited two-sheeted covering phi : <(<Delta>)over tilde>(0) --> Delta (0) with the branch points {phi (-1)(c(n))}, set (R) over tilde = phi (-1)(R). In the case c(n) = 2(-n), it was proved that if a uniqueness theorem is valid for H-infinity(R) at z = 0, then the Myrberg phenomenon H-infinity(R) o phi = H-infinity((R) over tilde) occurs. One might suspect that the converse also holds. In this paper, contrary to this intuition, we show that the converse of this previous result is not true. In addition, we generalize the previous result for more general sequences {c(n)}. By this generalization we can even partly simplify the previous proof.