On a Class of Functions Defined by Takahashi and Nunokawa

摘要

Let A denote the class of analytic functions f(z) in the unit disc U = {z: |z| < 1} normalized so that f(0) = f′(0) - 1 = 0. Recently Takahashi and Nunokawa in their work: Takahashi, N.; Nunokawa, M. A certain connection between starlike and convex functions. Appl. Math. Lett. 16 (2003), no. 5, 653-655, introduced the following subclass of the class of starlike functions STS(μ_1,μ_2) = {f ∈ A: (πμ_1)/2 < arg (zf'(z))/(f(z)) < (πμ_2)/2, z ∈ U}, where — 1 ≤ μ_1 < μ_2 ≤ 1. Here this class is studied further by methods from the theory of differential subordinations and some simple sufficient conditions that imply A C STS(μ_1, μ_2) are given. Also, comparison with some classical results is done.