We consider functions f analytic in the unit disc and assume the power series representation of the form f ( z ) = z + a n + 1 z n + 1 + a n + 2 z n + 2 + … where a n + 1 is fixed throughout. We provide a unified approach to radius convexity problems for different subclasses of univalent analytic functions. Numerous earlier estimates concerning the radius of convexity such as those involving fixed second coefficient, n initial gaps, n + 1 symmetric gaps, etc. are discussed. It is shown that several known results, follow as special cases of those presented in this paper.