Meromorphic univalent function with negative coefficient

Abstract

Let M n be the classes of regular functions f ( z ) = z − 1 + a 0 + a 1 z + … defined in the annulus 0 < | z | < 1 and satisfying Re I n + 1 f ( z ) I n + 1 f ( z ) > 0 , ( n ∈ ℕ 0 ) , where I 0 f ( z ) = f ( z ) , I f ( z ) = ( z − 1 − z ( z − 1 ) − 2 ) ∗ f ( z ) , I n f ( z ) = I ( I n − 1 f ( z ) ) , and ∗ is the Hadamard convolution. We denote by Γ n = M n ⋃ Γ , where Γ denotes the class of functions of the form f ( z ) = z − 1 + ∑ k = 1 ∞ | a k | z k . We obtained that relates the modulus of the coefficients to starlikeness for the classes M n and Γ n , and coefficient inequalities for the classes Γ n .