抄録
An analytic function f (z) = z + a2z2 + ··· in the unit disk Δ = {z : |z| < 1} is said to be in $¥mathcal{U}$ (λ, μ) if $$ ¥left|f'(z)¥left(¥frac{z}{f(z)} ¥right)^ {¥mu +1}-1 ¥right| ¥le ¥lambda ¥quad (|z|<1) $$ for some λ ≥ 0 and μ > -1. For -1 ≤ α ≤ 1, we introduce a geometrically motivated $¥mathcal{S}$p (α)-class defined by $${¥mathcal S}_p(¥alpha) = ¥left ¥{f¥in {¥mathcal S}: ¥left |¥frac{zf'(z)}{f(z)} -1¥right |¥leq {¥rm Re} ¥frac{zf'(z)}{f(z)}-¥alpha, ¥quad z¥in ¥Delta ¥right ¥},$$ where ${¥mathcal S}$ represents the class of all normalized univalent functions in Δ. In this paper, the authors determine necessary and sufficient coefficient conditions for certain class of functions to be in $¥mathcal{S}$p(α). Also, radius properties are considered for $¥mathcal{S}$p (α)-class in the class $¥mathcal{S}$. In addition, we also find disks |z| < r : = r (λ, μ) for which $¥frac{1}{r}$ f (rz) ∈ $¥mathcal{U}$ (λ, μ) whenever f ∈ $¥mathcal{S}$. In addition to a number of new results, we also present several new sufficient conditions for f to be in the class $¥mathcal{U}$ (λ, μ).
