Let P [ A , B ] , − 1 ≤ B < A ≤ 1 , be the class of functions p such that p ( z ) is subordinate to 1 + A z 1 + B z . A function f , analytic in the unit disk E is said to belong to the class K β * [ A , B ] if, and only if, there exists a function g with z g ′ ( z ) g ( z ) ∈ P [ A , B ] such that \beta $"> Re ( z f ′ ( z ) ) ′ g ′ ( z ) > β , 0 ≤ β < 1 and z ∈ E . The functions in this class are close-to-convex and hence univalent. We study its relationship with some of the other subclasses of univalent functions. Some radius problems are also solved.