Let S~* be the familiar class of normalized univalent functions in the unit disk.In [9], Keogh and Merkes proved that for a function f(z) = z +∑k=2∞ a_kz^k in the class S~*,then |a_3-λa_2~2| ≤ max{1, |3-4λ|}, λ∈ C. In this article, we investigate the corresponding problem for the subclass of starlike mappings defined on the unit ball in a complex Banach space, on the unit polydisk in Cnand the bounded starlike circular domain in C~■, respectively.
Let K be the familiar class of normalized convex functions in the unit disk. Keogh and Merkes proved the well-known result that maxf∈A ｜a3 - λa22｜ ≤ max{1/3, ｜λ - 1}, ,λ ∈ C, and the estimate is sharp for each ∈. We investigate the corresponding problem for a subclass of quasi-convex mappings of type B defined on the unit ball in a complex Banach space or on the unit polydisk in Cn. The proofs of these results use some restrictive assumptions, which in the case of one complex variable are automatically satisfied.
In this paper, we establish the Fekete and Szego inequality for a class of holomorphic functions in the unit disk, and then we extend this result to a class of holomorphic mappings on the unit ball in a complex Banach space or on the unit polydisk in Cn.
Let Sα*be the familiar class of normalized starlike functions of order α in the unit disk. In this paper, we establish the Fekete and Szeg? inequality for the class Sα*, and then we generalize this result to the unit ball in a complex Banach space or on the unit polydisk in Cn.
