2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/112609A new elementary proof for a theorem of D. Burns and S. Krantz on the rigidity of the analytic self maps of the unit disc was recently discovered by L. Baracco, D. Zaitsev, and G. Zampieri. We use their argument to generalize Burns-Krantz theorems on the unit disc and on the unit ball of ${\mathbb C}^n$.elsart, 6 pages, minor errors correctedComplex Variables30C80; 32A40text