Abstract
For an analytic function f (z) on the unit disk {pipe}z{pipe} < 1 with f (0) = f′(0) - 1 = 0 and f (z) ≠ 0, 0 < {pipe}z{pipe} < 1, we consider the power deformation f c(z) = z(f (z)/z) c for a complex number c. We determine those values c for which the operator → f c maps a specified class of univalent functions into the class of univalent functions. A little surprisingly, we will see that the set is described by the variability region of the quantity zf′(z)/ f (z), {pipe}z{pipe} < 1, for most of the classes that we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spirallike functions.
Original language | English |
---|---|
Pages (from-to) | 231-240 |
Number of pages | 10 |
Journal | Monatshefte fur Mathematik |
Volume | 167 |
Issue number | 2 |
DOIs | |
Publication status | Published - 2012 Aug |
Keywords
- Spirallike function
- Univalent function
- Variability region
ASJC Scopus subject areas
- Mathematics(all)