This thesis studies geometric and analytic properties of complex-valued analytic functions and logharmonic mappings in the open unit disk D. It investigates four
research problems. As a precursor to the first, let U be the class consisting of normalized analytic functions f satisfying |(z= f (z))2 f ′(z)−1| 0; or | f (z)=g(z)−1| < 1 in D; for g belonging to a certain class of
analytic functions. In most instances, the exact U -radius are found. A recent conjecture by Obradovi´c and Ponnusamy concerning the radius of univalence for a product involving univalent functions is also shown to hold true. The second problem deals with the Hankel determinant of analytic functions. For a normalized analytic function f ; let z f ′(z)= f (z) or 1+z f ′′(z)= f ′(z) be subordinate to a given analytic function
φ in D. Further let F be its kth-root transform, that is, F(z) = z[f(zk)=zk]1
