Integral representations of bounded starlike functions

For α ≥ 0 let ${\displaystyle {ℱ}_{\alpha }}$ denote the class of functions defined for |z| < 1 by integrating ${\displaystyle \frac{1}{{(1-xz)}^{\alpha }}}$ if α > 0, and log(1/(1-xz)) if α = 0, against a complex measure on |x| = 1. We study families of starlike functions where zf'(z)/f(z) ranges over a parabola with given focus and vertex. We prove a number of properties of these functions, among others that they are bounded and that they belong to ${\displaystyle {ℱ}_0}$. In general, it is only known that bounded starlike functions belong to ${\displaystyle {ℱ}_{\alpha }}$ for α > 0.