In the paper, by making use of the principle of subordination between analytic functions with the multiplier transforms defined by generalized Mittag-Leffler function, the authors investigate subclasses of univalent analytic functions, such as starlike functions, convex functions, close-to-convex functions and quasiconvex functions. Several inclusion relationships, inequality properties, subordination and superordination results associated with the multiplier transforms are proved and the sandwich-type results are also obtained.