Inequalities for the Coefficients of Meromorphic Starlike Functions with Nonzero Pole

In this article we consider functions f that are meromorphic and univalent in the unit disc ${\mathbb{D}}$ with pole at the point ${z = p \in (0, 1)}$ and having a Taylor expansion at the origin of the form $$f(z) = z + \sum _{n=2}^{\infty}a_n(f)z^n, \quad |z| < p.$$ The class of functions that satisfy the above conditions and map the unit disc such that ${\overline{\mathbb{C}} \setminus f(\mathbb{D})}$ is starlike with respect to a point w ₀ ( ≠ 0, ∞) will be denoted by Σ *(p, w ₀). We generalize and sharpen an inequality for a ₂(f), ${f \in \Sigma^*(p,w_0)}$ , proved by Miller (Proc Am Math Soc 80:607–613, 1980) by use of the coefficients a _n (f), n ≥ 3.