Variability regions of close-to-convex functions

M. Biernacki gave in 1936 concrete forms of the variability regions of z/f(z) and zf′(z)/f(z) of close-to-convex functions f for a fixed z with |z|< 1. The forms are, however, not necessarily convenient to determine the shape of the full variability region of zf′(z)/f(z) over all close-to-convex functions f and all points z with |z|<1. We propose a couple of other forms of the variability regions and see that the full variability region of zf′(z)/f(z) is indeed the complex plane minus the origin. We also apply them to study the variability regions of log[z/f(z)] and log[zf′(z)/f(z)].