On coefficient means of certain subclasses of univalent functions

Abstract:

Let $\mathcal {R}$ denote the class of regular functions whose derivatives have positive real part in the unit disc $\gamma$ and let $\mathcal {S}$ denote the class of functions starlike in $\gamma$. In this paper we investigate the rates of growth of the means ${s_n}(\lambda ) = {n^{ - 1}}\Sigma _1^n|{a_k}{|^\lambda }(0 < \lambda \leq 1)$ and ${t_n}(\lambda ) = {n^{ - 1}}\Sigma _1^n{k^\lambda }|{a_k}{|^\lambda }\;(\lambda > 0)$ as $n \to + \infty$ for bounded $f(z) = \Sigma _1^\infty {a_k}{z^k} \in \mathcal {R} \cup \mathcal {S}$. It is proved, for example, that the estimate ${t_n}(\lambda ) = o(1){(\log n)^{ - \alpha (\lambda )}}(n \to + \infty )$, where $\alpha (\lambda ) = \lambda /2$ for $0 < \lambda < 2$ and $\alpha (\lambda ) = 1$ for $\lambda \geq 2$, holds for such functions f, and that it is best possible for each fixed $\lambda > 0$ within the class $\mathcal {R}$ and for each fixed $\lambda \geq 2$ within the class $\mathcal {S}$. It is also shown that the inequality ${s_n}(1) = o(1){n^{ - 1}}{(\log n)^{1/2}}$, which holds for all bounded univalent functions, cannot be improved for bounded $f \in \mathcal {R}$. The behavior of ${t_n}(\lambda )$ as $n \to + \infty$ when ${a_k} \geq 0(k \geq 1)$ and $\lambda \geq 1$ is also examined.