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For α ≥ 0 let ${ℱ}_{\alpha }$ denote the class of functions defined for |z| < 1 by integrating $1/{(1-xz)}^{\alpha }$ 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 ${ℱ}_0$. In general, it is only known that bounded starlike functions belong to ${ℱ}_{\alpha }$ for α > 0.