For $\beta< 1$, n = 0, 1, 2, . . ., and $-\pi <\alpha\leq\pi$, we let
$M_n(\alpha,\beta)$ denote the family of functions $f(z) = z +\ldots$
that are analytic in the unit disk and satisfy there the condition
$Re\{(D^n f)'+\frac{1+e^{i\alpha}}{2(n+1)}z(D^n f)''\}>\beta$,
where $D^n f(z)$ is the Hadamard product or convolution of f with
$z/(1 − z){n+1}$. We prove the inclusion relations
$M_{n+1}(\alpha,\beta) \subset M_n(\alpha,\beta$,
and $M_n(\alpha,\beta) < M_n(\pi,\beta), \beta < 1$.
Extreme points, as well as integral and convolution characterizations, are found.
This leads to coefficient bounds and other extremal properties.
The special cases $M_0(\alpha,0)\equiv \mathcal{L}_\alpha$,
$M_n(\pi,\beta)\equiv M_n(\beta)$ have previously
been studied [16], [1].
