About an integral operator preserving meromorphic starlike functions

Let U = {z ∈ C : |z| 0 in U Let denote by Σk the class of starlike functions inΣk and by An the class of holomorphic functions g of the form: g(z) = z + an+1z + · · · , z ∈ U , n ≥ 1 With suitable conditions on k, p ∈ N,on c ∈ R,on γ ∈ C and on the function g ∈ Ak+1, the author shows that the integral operator Lg,c,γ : Σ→ Σ defined by: Kg,c(f)(z) ≡ c gc+1(z) ∫ z 0 f(t)g(t)e p dt, z ∈ U , f ∈ Σ maps Σk into Σ ∗ l , where l = min{p− 1, k}. ∗The author aknowleges support received from the“Conference of the German Academies of Sciences”( Konferenz der Deutschen Akademien der Wissenschaften ), with funds provided by the “Volkswagen Stiftung”.This work was done while the author was visiting the University of Hagen in Germany. 1991 Mathematics Subject Classification : Primary : 30C80,30C45, Secondary : 30D.