Certain classes of meromorphically multivalent functions with fixed argument of coefficients (Q2719888)

Let \(\Sigma_p\) denote the class of functions \(f\) of the form \(f(z)=z^{-p}+\sum_{n=1}^{\infty}a_nz^{n}\) \((p\in\mathbb{N}=\{1,2,\ldots \})\) that are analytic in the punctured unit disc \(\Delta \setminus \{0\}\), \(\Delta=\{z: |z|<1\}\). A function \(f\in \Sigma_p\) (with \(\arg a_n=\theta\) for \(n\in\mathbb{N}\)) is said to belong to the class \(\Sigma_p^{\theta}(A,B)\) for some \(0<B\leq 1\) and \(-B\leq A<B\) (\(B\neq 1\) or \(\cos\theta>0\)), \(f\) satisfies the condition \(-z^{p+1}f'(z)\prec p({ {1+Az} \over {1+Bz}})\), \( z\in \Delta.\) In this paper the authors have obtained coefficient estimates, distortion theorems and extreme points for the functions in \(\Sigma_p^{\theta}(A,B)\).