Coefficient inequality for a generalized subclass of analytic functions

Abstract

For α (0 ≤ α < 1), ɳ (0 ≤ ɳ ≤1), β (0 < β ≤ 1) , let H_λ,μ^m(α; β, ɳ) be the class of normalised functions defined in the unit disk U by

R (1 − ɳ)z(D_λ,μ^m f(z))'+ɳz(D_λ,μ^m+1 f(z))'/(1 − ɳ)D_λ,μ^mg(z)+ɳ(D_λ,μ^m+1g(z)> 0             (1)

where D_λ,μ^m is a linear multiplier differential operator and g P_λ,μ^m (α; β, ɳ) is the class of normalized functions defined in the unit disk U by

|arg(1 − ɳ)z(D_λ,μ^m f(z))'+ɳz(D_λ,μ^m+1 f(z))'/(1 − ɳ)D_λ,μ^mf(z)+ɳ(D_λ,μ^m+1 f(z) - α) < π/2  (2)

f∈ H_λ,μ^m(α; β, ɳ) and given by f(z)=z+a₂z²+a₃z³+..., a sharp upper bound is obtained for |a₃-ξa₂²| when A₂ξA₁² where A₁ and A₂ are given below.