Growth properties of the modular function J

Abstract The 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 ⩾ 1 2 , ¦j(z)¦ increases when Re z varies from − 1 2 to 0 and decreases when Re z varies from 0 to 1 2 ; for Re z fixed, Re z ϵ [− 1 2 , + 1 2 ], ¦j(z)¦ increases when Im z varies from (1 − ( Re z) 2 ) 1 2 , to + ∞ .