On the derivatives of a family of analytic functions

Abstract

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].