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

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Ponnusamy S., Vasudevarao A., Vuorinen M.

Colloquium Mathematicum

2009

tom 116

nr 1

31-46

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 𝔻 with f(0) = 1 and $Re(2π/μ zf'(z)/f(z) + (1+z)/(1-z)) > 0$ in 𝔻. For any fixed z₀ in the unit disk, a ∈ ℂ with |a| ≤ 1 and λ ∈ 𝔻̅, we shall determine the region of variability V(z₀,λ) for log f(z₀) when f ranges over the class $ℱ_{μ}(λ) = {f ∈ ℱ_{μ}: f'(0) = (μ/π)(λ - 1) and f''(0) = (μ/π)(a(1-|λ|²) + (μ/π)(λ-1)² - (1-λ²))}$. In the final section we graphically illustrate the region of variability for several sets of parameters.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Ponnusamy S.

1 Vasudevarao A.

1 Vuorinen M.

1 2009

Wyniki wyszukiwania

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