Criteria for strongly starlike functions

Let A n be the family of normalized regular functions f, where f(z)=z+an+1zn+1+…, in the unit disc Δ and let S(α) and T λ be the families of functions f such that respectively. S(α) is the class of strongly starlike functions. Further, let S ∗(α) denote the well-known family of starlike functions of order α. The purpose of this paper is to find conditions so that 1. satisfying satisfying . 2. satisfying is in S(α) or in 3. satisfying is in S(α) or in S ∗(α); 4. satisfying Re f′(z)≥1−λ1,|g′(z)−1|≤λ2, imply z(f∗g)′ is in S(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.