Let denote the family of normalized regular functions defined in the unit disc . For and real , let denote the family of functions such that for . Given a generalized second order polylogarithm function
we place conditions on the parameters , and to guarantee that the Hadamard product of the power series will be univalent, starlike or convex. We give conditions on and to describe the geometric nature of the function . We note that for , the function satisfies the differential equation
and has the integral representation
If , and with , we see that reduces to the well-known Bernardi transform. If and , is the operator considered by R. Ali and V. Singh [Complex Variables, Theory Appl. 26, No. 4, 299-309 (1995; Zbl 0851.30005)] with an additional assumption that . Thus, the operator is the natural choice to study its behaviour. By making in the class of convex functions, we also find a sufficient condition for to belong to the class . Several open problems have been raised at the end.