The Fekete-Szegö Inequality and Successive Coefficients Difference for a Subclass of Close-to-Starlike Mappings in Complex Banach Spaces

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

“…Using the above equalities, (3)(4)(5)(6)(7)(8) and (3)(4)(5)(6)(7)(8)(9), we obtain all of the desired conclusions about Theorems 3.1. The example which shows the sharpness of Theorem 3.1 is the same as the mapping defined in [37]. This completes the proof of Theorem 3.1.…”

“…In this section, by using the proof methods different from those appeared in [39] and [37], we obtain the corresponding results of norm type and functional type for subclasses of close-to-quasiconvex mappings of type B and close-to-starlike mappings defined on the open unit ball in a complex Banach space (see Theorem 1.7).…”

“…More recently, Xu et al [37] gave another extension of Theorem 1.5 to higher dimensions, and established the following Fekete and Szegő inequality for the subclass of close-to-starlike mappings on the open unit ball ‫ނ‬ in a complex Banach space with respect to H ∈ S * ‫. )ނ(‬ Theorem 1.8 [37].…”