Starlikeness criteria for a certain class of analytic functions

We denote by AA, the class of all analytic functions ff in the unit disc Δ={z∈C:|z|<1}Δ={z∈C:|z|<1} with the normalization f(0)=f′(0)−1=0f(0)=f′(0)−1=0. For a positive number λ>0λ>0, we denote by U3(λ)U3(λ) the class of all f(z)=z+∑n=2∞anzn∈A, such that a3−a22=0, and satisfying the condition |(zf(z))2f′(z)−1|<λ,z∈Δ. A function f∈Af∈A is said to be in SR(γ)SR(γ) if |argf′(z)|<πγ/2|argf′(z)|<πγ/2. In this paper, we find conditions on λλ, αα and γγ such that U3(λ)U3(λ) is included in the class of all starlike functions of order αα, or the class of all strongly starlike functions of order γγ, or SR(γ)SR(γ), respectively.