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In 2005, Ponnusamy and Sahoo have introduced a special subclass of univalent functions Un(\lambda) (n 2 N, > 0) and obtained some geometrical properties, including strongly starlikeness and convexity, for the functions of this subclass Un(). Moreover, they have studied some important properties of an integral transform connected with these subclasses. The aim of the present paper is to investigate another important concept of majorization for the functions belonging to the class Un(\lambda) (0 <\lambda=< 1). We shall also discuss a majorization problem for some special integral transforms.