Note on the Fekete–Szegö Problem for Spirallike Mappings in Banach Spaces

Abstract

In this paper we introduce the Fekete-Szegö type mapping in the open unit ball of a complex Banach space. All previously studied modifications of the Fekete-Szegö functional are either special cases or 'components' of the mapping we introduce.The study involves the examination of transforms of the Fekete-Szegö mapping under specific transformations applied to given holomorphic mappings.We show that for a mapping f , the third order Fréshet derivative of the inverse mapping f −1 and of elements of the semigroup generated by f can be expressed in terms of the Fekete-Szegö mapping. Estimates of the Fekete-Szegö mapping over some subclasses of semigroup generators and of starlike mappings are also presented.

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].…”

“…By using (3-10), (3)(4)(5)(6)(7)(8)(9)(10)(11) and (3)(4)(5)(6)(7)(8)(9)(10)(11)(12), we obtain k ∈ S * . Let s(x) = (D F(x)) −1 H (x), and let…”

On refinements of the Fekete-Szegö inequality for a class of holomorphic mappings associated with spirallike mappings

“…Remark 3.4. We remark that Theorems 3.2-3.3 are proved under weaker assumptions than in the above-cited works ( [3], [11]).…”

“…More recently, Elin and Jacobzon [3] generalized Theorem 1.6 to the unit ball of a complex Banach space as follows.…”

The Fekete-Szego problem for spirallike mappings and non-linear resolvents in Banach spaces

“…Proof. The first assertion is evident (for detailed calculation see [7]). To prove the second one we note that there is a function…”