Region of variability for spiral-like functions with respect to a boundary point

S. Ponnusamy, A. Vasudevarao, and M. Vuorinen. "Region of variability for spiral-like functions with respect to a boundary point." Colloquium Mathematicae 116.1 (2009): 31-46. <http://eudml.org/doc/283869>.

@article{S2009,
abstract = {For μ ∈ ℂ such that Re μ > 0 let $ℱ_\{μ\}$ denote the class of all non-vanishing analytic functions f in the unit disk with f(0) = 1 and $Re(2π/μ zf^\{\prime \}(z)/f(z) + (1+z)/(1-z)) > 0$ in . For any fixed z₀ in the unit disk, a ∈ ℂ with |a| ≤ 1 and λ ∈ ̅, we shall determine the region of variability V(z₀,λ) for log f(z₀) when f ranges over the class $ℱ_\{μ\}(λ) = \{f ∈ ℱ_\{μ\}: f^\{\prime \}(0) = (μ/π)(λ - 1) and f^\{\prime \prime \}(0) = (μ/π)(a(1-|λ|²) + (μ/π)(λ-1)² - (1-λ²))\}$. In the final section we graphically illustrate the region of variability for several sets of parameters.},
author = {S. Ponnusamy, A. Vasudevarao, M. Vuorinen},
journal = {Colloquium Mathematicae},
keywords = {starlike function; spiral-like function with respect to a boundary point},
language = {eng},
number = {1},
pages = {31-46},
title = {Region of variability for spiral-like functions with respect to a boundary point},
url = {http://eudml.org/doc/283869},
volume = {116},
year = {2009},
}

TY - JOUR
AU - S. Ponnusamy
AU - A. Vasudevarao
AU - M. Vuorinen
TI - Region of variability for spiral-like functions with respect to a boundary point
JO - Colloquium Mathematicae
PY - 2009
VL - 116
IS - 1
SP - 31
EP - 46
AB - For μ ∈ ℂ such that Re μ > 0 let $ℱ_{μ}$ denote the class of all non-vanishing analytic functions f in the unit disk with f(0) = 1 and $Re(2π/μ zf^{\prime }(z)/f(z) + (1+z)/(1-z)) > 0$ in . For any fixed z₀ in the unit disk, a ∈ ℂ with |a| ≤ 1 and λ ∈ ̅, we shall determine the region of variability V(z₀,λ) for log f(z₀) when f ranges over the class $ℱ_{μ}(λ) = {f ∈ ℱ_{μ}: f^{\prime }(0) = (μ/π)(λ - 1) and f^{\prime \prime }(0) = (μ/π)(a(1-|λ|²) + (μ/π)(λ-1)² - (1-λ²))}$. In the final section we graphically illustrate the region of variability for several sets of parameters.
LA - eng
KW - starlike function; spiral-like function with respect to a boundary point
UR - http://eudml.org/doc/283869
ER -