Meromorphic univalent function with negative coefficient

Abstract

Let Mn be the classes of regular functions f(z) = z−1 + a0 + a1z + … defined in the annulus 0 < |z| < 1 and satisfying , (n ∈ ℕ0), where I0f(z) = f(z), If(z) = (z−1 − z(z−1)−2)∗f(z), Inf(z) = I(In−1f(z)), and ∗ is the Hadamard convolution. We denote by Γn = Mn ⋃ Γ, where Γ denotes the class of functions of the form . We obtained that relates the modulus of the coefficients to starlikeness for the classes Mn and Γn, and coefficient inequalities for the classes Γn.