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M. Obradovic, S. Ponnusamy

Radius properties for subclasses of univalent functions

Keywords: coefficient inequality, analytic, univalent, close-to-convex, starlike and convex functions

A normalized analytic function f(z) = z + a₂z² + ··· (|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) ≠⊆ U ≠⊆ 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.

We also show that if f ∈ S, then one has r⁻¹f(rz) ∈ P(2) for 0 < r ≤ r₀, where r₀ = 0.60629, correctly rounded to six decimal places, is the unique root of the equation 2r⁸ − 9r⁶ + 10r⁴ − 8r² + 2 = 0.

Analysis, Oldenbourg Wissenschaftsverlag

Print ISSN: 0174-4747
Volume: 25, 03/2005
Pages: 183 - 188

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