Region of variability for certain classes of univalent functions satisfying differential inequalities

: Ponnusamy S., Vasudevarao A., Vuorinen M.

: 2009

: Complex Variables and Elliptic Equations

: 54

: 10

: 899

: 922

: 24

: 1747-6933

DOI: https://doi.org/10.1080/17476930802657616

: http://api.elsevier.com/content/abstract/scopus_id:75249100943

For complex numbers α, β and M ∈ ℝ with 0 < M ≤ {pipe}α{pipe} and {pipe}β{pipe} ≤ 1, let B(α, β, M) be the class of analytic and univalent functions f in the unit disk D with f(0) = 0, f′(0) = α and f″(0) = Mβ satisfying {pipe}zf″(z){pipe} ≤ M, z ∈ D. Let P(α, M) be the another class of analytic and univalent functions in D with f(0) = 0, f′(0) = α satisfying Re(zf″(z)) > -M, z ∈ D, where α ∈ ℂ\{0}, 0 < M ≤ 1/log 4. For any fixed z ∈ D, and λ ∈ D̄ we shall determine the region of variability V (j = 1, 2) for f′(z) when f ranges over the class S (j = 1, 2), where In the final section we graphically illustrate the region of variability for several sets of parameters. © 2009 Taylor & Francis.