LetAnbe the family of normalized regular functions f, where f(z)=z+an+1zn+1+…, in the unit disc Δ and letS(α) andTλbe the families of functionsfsuch thatrespectively.S(α) is the class of strongly starlike functions. Further, letS*(α) denote the well-known family of starlike functions of order α. The purpose of this paper is to find conditions so that 1.satisfyingsatisfying. 2.satisfyingis inS(α) or in3.satisfyingis inS(α) or inS*(α); 4.satisfying Re f′(z)≥1−λ1,|g′(z)−1|≤λ2, implyz(f*g)′ is inS(α). The results of this paper extend the previously known results and improve some criteria for starlike functions. These are then used to obtain new information for uniformly convex functions and Bemardi integral transform.