In this paper we present and discuss di↵erent types of Ulam stability: Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability for Cauchy diferential equation of fractional order in the unit disk. Moreover, the generalized Ulam -Hyers stability for univalent solution is introduced and the existence and uniqueness of boundary problem are established.