  For given non-negative real numbers $\alpha_k$ with $ \sum_{k=1}^{m}\alpha_k
=1$ and normalized analytic functions $f_k$, $k=1,\dotsc,m$, defined on the
open unit disc, let the functions $F$ and $F_n$ be defined by $
F(z):=\sum_{k=1}^{m}\alpha_k f_k (z)$, and $F_{n}(z):=n^{-1}\sum_{j=0}^{n-1}
e^{-2j\pi i/n} F(e^{2j\pi i/n} z)$. This paper studies the functions $f_k$
satisfying the subordination $zf'_{k} (z)/F_{n} (z) \prec h(z)$ 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 connection with the previous works are
indicated.
