About an integral operator preserving meromorphic starlike functions

Let U = {z is an element of C : \z\ < 1} be the unit disc in the complex plane. Let Sigma(k) be the class of meromorphic functions f in U having the form: f(z) = 1/z + alpha(k)z(k) + ..., 0 < \z\ < 1, k greater than or equal to 0 A function f is an element of Sigma = Sigma(0) is called starlike if Re [-zf'(z)/f(z)] > 0 in U Let denote by Sigma(k)* the class of starlike functions in Sigma(k) and by A(n) the class of holomorphic functions g of the form: g(z) = z + a(n+1)z(n+1) + ..., z is an element of U, n greater than or equal to 1 With suitable conditions on k, p is an element of N, on c is an element of R, on gamma is an element of C and on the function g is an element of A(k+1), the author shows that the integral operator L-g,L-c,L-gamma: Sigma --> Sigma defined by: K-g,K-c(f)(z) = c/g(c+1)(z) integral(0)(z) f(t)g(c)(t)e(gamma tP) dt, z is an element of U, f is an element of Sigma maps Sigma(k)* into Sigma(l)*, where l = min{p-1, k}.