Region of variability of univalent functions f(z) for which zf′(z) is spirallike

For a complex number α with Re α > 0 let S α(λ) be the class of analytic functions f in the unit disk D with f(0) = 0 = f′(0) - 1, f″(0) = 2λe -1α cos α satisfying Reeiα (1 + f′(z)/zf″(z)) > 0 for z ∈ D. For z 0 ∈ D fixed, we determine the region of variability for log f′(z 0) when f ranges over the class Sα (λ). As a consequence, we obtain an estimate for a pre-Schwarzian norm for Sα(0). © 2008 University of Houston.