A family of accelerated iterative methods for the simultaneous approximation of complex zeros of a class of analytic functions is proposed. Considered analytic functions have only simple zeros inside a simple smooth closed contour in the complex plane. It is shown that the order of convergence of the basic family can be increased from four to five and six using Newton's and Halley's corrections, respectively. The improved convergence is achieved on the account of additional calculations of low computational cost, which significantly increases the computational efficiency of the accelerated methods. Numerical examples demonstrate a good convergence properties, fitting very well theoretical results.