Radius properties for subclasses of univalent functions

摘要

A normalized analytic function f(z) = z + a_2z~2 + … (|z| < 1) is said to be in U (resp. P(2)) if for |z| < 1, |f′(z) (z/(f(z))~2-1| ≤ 1 (resp. |(z/(f(z))″| ≤ 2). It is known that P(2) is contained in U is contained in S, where S denotes the set of all normalized analytic functions that are univalent in |z| < 1. In this paper, we prove a general result which implies that sup {r > 0 : r~(-1) f(rz) ∈ U for every f ∈ S} = 1/2~(1/2). We also show that if f ∈ S, then one has r~(-1) f(rz) ∈ P(2) for 0 < r ≤ r_0, where r_0 = 0.60629, correctly rounded to six decimal places, is the unique root of the equation 2r~8 - 9r~6 + 10r~4 -8r~2 + 2 = 0.