2011/08/02 by Takao Kato, Toshiyuki Sugawa, Kato, Takao +3
Mathematics · #30C45 (Primary) 30C75 (Secondary) #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:30C45 #msc:30C75
paper · pdf · doi:10.48550/arxiv.1108.0461
12 pages
arxiv created 2012/09/24 · arxiv updated 2012/09/25
M. Biernacki gave 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 in 1936. 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 will 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)].