Geometric Properties of Some Generalized Mathieu Power Series inside the Unit Disk

We consider two parametric families of special functions: One is defined by a power series generalizing the classical Mathieu series, and the other one is a generalized Mathieu type power series involving factorials in its coefficients. Using criteria due to Fejer and Ozaki, we provide sufficient conditions for these functions to be close-to-convex or starlike inside the unit disk, and thus univalent. One of our proofs is assisted by symbolic computation.