COEFFICIENT ESTIMATES FOR CERTAIN SUBCLASSES OF ANALYTIC FUNCTIONS OF COMPLEX ORDER

Abstract

In this paper, we introduce and investigate each of the following subclasses: $$\mathcal{S}_g(\lambda, \gamma)\,\,\,\, \text{and} \,\,\,\, \mathcal{K}_g(\lambda, \gamma, m; u) \,\,\,\, \Big(0\!\leqq\! \lambda\! \leqq\! 1; u\!\in\! \mathbb{R}\setminus (-\infty, -1]; \ m\in \mathbb{N}\setminus\{1\}\Big) $$ of analytic functions of complex order $\gamma \in \mathbb{C} \setminus \{0\}$, $g: \mathbb{U} \rightarrow \mathbb{C}$ being some suitably constrained convex function in the open unit disk $\mathbb{U}$. We obtain coefficient bounds and coefficient estimates involving the Taylor-Maclaurin coefficients of the function $f(z)$ when $f(z)$ is in the class $\mathcal{S}_g(\lambda,\gamma)$ or in the class $\mathcal{K}_g(\lambda,\gamma,m;u)$. The various results, which are presented in this paper, would generalize and improve those in related works of several earlier authors.