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Starlikoness of Certain Analytic Functions with Missing Coefficients
Author(s): 
Pages: 489-494
Year: Issue:  4
Journal: Journal of Suzhou University Natural Science

Keyword:  Ufnivalent functionsmissing cocfficientsstarlikeness.;
Abstract: <正> Let Sn denote class of functions f(z) analytic and univa-lent in the unit disc Δ={z: |z|<1}, with expansion f(z)=z+ an+1zn+1+an+2zn+2+..., n≥1. Let Sn*(λ) and Sn, (m,M) denote the subclasses of Sn. and satisfying the condi tions Re{zf’(z)/f(z)}>λ, z∈Δ, 0≤λ<1 and |zf’(z)/f(z)—m|<M, z∈Δ, |m—1|<M≤m respectivelh. Let Pn(μ) denote the class of functions of the form p(z)=1+cnzn+cn+1zn+1+…, n≥1, which are analytic and satisfy Rep(z)>μ, z∈Δ, 0≤μ<1. In this note we study starlikeness of functions f(z) defined bywhere F(z)∈Sn*(λ), g(z)∈Sn(m,M)(or g(z)/z∈Pn(δ), 0≤δ<1), h(z)∈Sn(m, M), p(z)∈Pn(μ), a>0, β≥0, γ≥0 and c+αλ≥ 0. Our results are sharp and extend some recent results due to Bhargava and Pandey.
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