For analytic functions f ( z ) = z p + a p + 1 z p + 1 + ⋯ in the open unit disk U and a polynomial Q ( z ) of degree 0$"> n > 0 , the function F ( z ) = f ( z ) [ Q ( z ) ] β / n is introduced. The object of the present paper is to determine the radius of p -valently strongly starlikeness of order γ for F ( z ) .