  In this article, we obtain existence and uniqueness results to some problems
involving complex nonlinear fractional differential equations (FDEs) in the
closed unit disc of C. By help of these results, we prove that some IVPs for
some fractional differential equations with Caputo or Riemann-Liouville
derivative admit at least one local (or unique) solution continuous on a closed
interval $[0,R]$ and real analytic on $(0,R),$ where $0<R\leq 1.$
