Distortion properties of $p$-fold symmetric alpha-starlike functions

Abstract:

Starlike functions $f$ which are of Mocanu type $\alpha$ and have power series of the form \[ f(z) = z + {a_{p + 1}}{z^{p + 1}} + {a_{2p + 1}}{z^{2p + 1}} + \cdots ,\] where $p = 1,2,3, \cdots$, are shown to satisfy the relation $f(z) = {[g({z^p})]^{1/p}}$ where $g$ is of Mocanu type $p\alpha$ with power series $g(z) = z + {b_2}{z^2} + {b_3}{z^3} + \cdots$. Distortion results dealing with the $\tfrac {1}{4}$-theorem and bounds on $|f(z)|$ are obtained.