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2 results

  • UNIVALENT FUNCTIONS AND RADIAL GROWTH

  • Part of
  • J. B. Twomey
  • Journal:
    Mathematika / Volume 61 / Issue 3 / September 2015
    Published online by Cambridge University Press:
    16 January 2015, pp. 531-546
    Print publication:
    September 2015

  • We address a question raised by Anderson, Hayman and Pommerenke relating to a classical result on univalent functions $f$ in the unit disc due to Spencer, and involving the size of the set of ${\it\theta}\in [-{\it\pi},{\it\pi}]$ for which we have $\log |f(r\text{e}^{\text{i}{\it\theta}})|\neq o(\log (1/(1-r)))$ as $r\rightarrow 1.$ An answer is given in terms of a certain generalized capacity, and also in terms of Hausdorff measure. Further results regarding the radial growth of univalent functions are also established, and some examples are constructed which relate to the sharpness of these results.