Starlikeness criteria for a certain class of analytic functions

We denote by A, the class of all analytic functions f in the unit disc Δ=z∈C:

z

<1 with the normalization f(0)=f′(0)-1=0. For a positive number λ>0, we denote by U3(λ) the class of all f(z)=z+∑n=2∞anzn∈A, such that a3-a 22=0, and satisfying the condition

(z/f(z))2f′(z)- 1

<λ,z∈Δ. A function f∈A is said to be in SR(γ) if

argf′(z)

<πγ2. In this paper, we find conditions on λ, α and γ such that U3(λ) is included in the class of all starlike functions of order α, or the class of all strongly starlike functions of order γ, or SR(γ), respectively. © 2010 Elsevier Ltd. All rights reserved.