ON FUNCTIONS STARLIKE WITH RESPECT TO n-PLY SYMMETRIC, CONJUGATE AND SYMMETRIC CONJUGATE POINTS

For given non-negative real numbers alpha(k) with Sigma(m) (k=1) alpha(k) = 1 and normalized analytic functions f(k), k = 1; : : :;m, defined on the open unit disc, let the functions F and Fn be defined by F(z) := Sigma(m)(k=1) alpha(k)f(k)(z), and Fn(z) := n(-1) Sigma (n-1) (j=0) (-2j pi i/n) F(e(2j pi i)/n(z)). This paper studies the functions fk satisfying the subordination zf '(k) (z)=Fn(z) (sic) h(z), where the function h is a convex univalent function with positive real part. We also consider the analogues of the classes of starlike functions with respect to symmetric, conjugate, and symmetric conjugate points. Inclusion and convolution results are proved for these and related classes. Our classes generalize several well-known classes and the connections with the previous works are indicated.