Author(s): Yang Dinggong Mathematics Department
Pages: 489-494
Year: 1988 Issue:  4
Journal: Journal of Suzhou University Natural Science
Keyword:  Ufnivalent functions; missing cocfficients; starlikeness.;
Abstract: <正> Let S_n denote class of functions f（z） analytic and univa-lent in the unit disc Δ={z: |z|<1}, with expansion f（z）=z+ a_n+1zⁿ⁺¹+a_n+2zⁿ⁺²+..., n≥1. Let S_n^*（λ） and S_n, （m,M） denote the subclasses of S_n. 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 P_n（μ） denote the class of functions of the form p（z）=1+c_nzⁿ+c_n+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）∈S_n^*（λ）, g（z）∈S_n（m,M）（or g（z）/z∈P_n（δ）, 0≤δ<1）, h（z）∈S_n（m, M）, p（z）∈P_n（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.