One parameter family of univalent biharmonic mappings

In this paper, we provide sufficient conditions to construct sensepreserving and univalent biharmonic mappings that arises from analytic functions which are not necessarily univalent in the unit disk |z| <1. Also we state and prove several theorems under different weaker hypothesis ineach case, leading to an affirmative answer to the radius problem posed by Y. Abu Muhanna in 2008.