Growth properties of the modular function j

AbstractThe modular function j is of great importance, specially in number theory. In this paper, we determine how ¦j(z)¦ varies when Im z or Re z remains fixed: for Im z fixed, Im z ⩾ 12, ¦j(z)¦ increases when Re z varies from −12 to 0 and decreases when Re z varies from 0 to 12; for Re z fixed, Re z ϵ [−12, + 12], ¦j(z)¦ increases when Im z varies from (1 − (Re z)2)12, to + ∞