On the coefficient inequalities for some classes of holomorphic mappings in complex Banach spaces

Qinghua Xu, Xiaohua Yang and Taishun Liu

Vol. 329 (2024), No. 1, 183–199

DOI: 10.2140/pjm.2024.329.183

Abstract

Let $\mathcal{𝒞}$ be the familiar class of normalized close-to-convex functions in the unit disk. Koepf (1987) proved that for a function $f(z)=z+{\sum <!--FUNCTION\; APPLICATION-->}_{k=2}^{\infty }{a}_k{z}^k$ in the class $\mathcal{𝒞}$,

$$|a_3-\lambda a_2^2|\le \{\begin{array}{cc}3-4\lambda ,\hfill & \lambda \in \left[0,\frac{1}{3}\right],\hfill \\ & \\ \frac{1}{3}+\frac{4}{9\lambda },\hfill & \lambda \in \left[\frac{1}{3},\frac{2}{3}\right],\hfill \\ & \\ 1,\hfill & \lambda \in \left[\frac{2}{3},1\right]\hfill \end{array}\text{ and}\left|\left|a_3\right|-\left|a_2\right|\right|\le 1.$$

Recently, Xu et al. (2023) generalized the above results to a subclass of close-to-quasiconvex mappings of type $B$ defined on the open unit polydisc in ${ℂ}^n$, and to a subclass of close-to-starlike mappings defined on the open unit ball of a complex Banach space, respectively. In the first part of this paper, by using different methods, we obtain the corresponding results of norm type and functional type on the open unit ball in a complex Banach space. We next give the coefficient inequalities for a subclass of $g$-starlike mappings of complex order $\lambda$ on the open unit ball of a complex Banach space, which generalize many known results. Moreover, the proofs presented here are simpler than those given in the related papers.

Keywords

Fekete and Szegő problem, close-to-quasiconvex mapping of type $B$, close-to-starlike mapping, $g$-starlike mapping of complex order $\lambda$

Mathematical Subject Classification

Primary: 32H02

Secondary: 30C55

Milestones

Received: 2 August 2023

Revised: 12 March 2024

Accepted: 11 May 2024

Published: 12 June 2024