2020/06/28 by Ali, Md Firoz, Allu, Vasudevarao, Yanagihara, Hiroshi
#30C45 #30C75 #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2006.15572
We continue our study on variability regions in \citeAli-Vasudevarao-Yanagihara-2018, where the authors determined the region of variability VΩj (z0, c ) = \ ∫0z0 zj(g(z)-g(0)) d z : g(\mathbb D) ⊂ Ω, (P-1 ∘ g) (z) = c0 +c1z + ⋯ + cn zn + ⋯ \ for each fixed z0 ∈ \mathbb D, j=-1,0,1,2, … and c = (c0, c1 , … , cn) ∈ ℂn+1, when Ω\subsetneqℂ is a convex domain, and P is a conformal map of the unit disk \mathbb D onto Ω. In the present article, we first show that in the case n=0, j=-1 and c=0, the result obtained in \citeAli-Vasudevarao-Yanagihara-2018 still holds when one assumes only that Ω is starlike with respect to P(0). Let CV(Ω) be the class of analytic functions f in \mathbb D with f(0)=f'(0)-1=0 satisfying 1+zf''(z)/f'(z) ∈ Ω. As applications we determine variability regions of log f'(z0) when f ranges over CV(Ω) with or without the conditions f''(0)= λ and f'''(0)= μ. Here λ and μ are arbitrarily preassigned values. By choosing particular Ω, we obtain the precise variability regions of log f'(z0) for other well-known subclasses of analytic and univalent functions.