Meromorphic univalent function with negative coefficient (Q1321875)

The author's abstract. Let \(M_ n\) be the classes of regular functions \(f(z) = z^{-1} + a_ 0 + a_ 1z + \cdots\) defined in the annulus \(0 < | z | < 1\) and satisfying \(\text{Re} {I^{n + 1} f(z) \over I^{n + 1} f(z)} > 0\), \((n \in \mathbb{N}_ 0)\), where \(I^ 0f(z) = f(z)\), \(If(z) = (z^{-1} - z(z - 1)^{-2})* f(z)\), \(I^ nf(z) =I(I^{n-1} f(z))\), and \(*\) is the Hadamard convolution. We denote by \(\Gamma_ n = M_ n \cup \Gamma\), where \(\Gamma\) denotes the class of functions of the form \(f(z) = z^{-1} + \sum^ \infty_{k = 1} | a_ k | z^ k\). We obtained a theorem that relates the modulus of the coefficients to starlikeness for the classes \(M_ n\) and \(\Gamma_ n\), and coefficient inequalities for the classes \(\Gamma_ n\).