A unified approach for univalent functions with negative coefficients using the Hadamard product

Abstract

For given analytic functions ϕ(z) = z + Σ ^∞ _n=2 λ _n z ⁿ, Ψ(z) = z + Σ ^∞ _n=2 μ with λ _n ≥ 0, μ _n ≥ 0, and λ _n ≥ μ _n and for α, β (0≤α<1, 0<β≤1), let E(φ,ψ; α, β) be of analytic functions ƒ(z) = z + Σ ^∞ _n=2 a _n z ⁿ in U such that f(z)*ψ(z)≠0 and

$$\left| {(f(z)*\varphi (z))/((f(z)*\psi (z)) - 1\left| { < \beta } \right|(f(z)*\varphi (z))/((f(z)*\psi (z)) + (1 - 2\alpha )} \right|$$

for z∈U; here, * denotes the Hadamard product. Let T be the class of functions ƒ(z) = z - Σ ^∞ _n=2 |a _n | that are analytic and univalent in U, and let E _T (φ,ψ;α,β)=E(φ,ψ;α,β)∩T. Coefficient estimates, extreme points, distortion properties, etc. are determined for the class E _T (φ,ψ;α,β) in the case where the second coefficient is fixed. The results thus obtained, for particular choices of φ(z) and ψ(z), not only generalize various known results but also give rise to several new results.