Unified solution of Fekete-Szegö problem for subclasses of starlike mappings in several complex variables

Abstract

Let $\begin{array}{} \mathcal {S}^*_\psi \end{array}$ be a subclass of starlike functions in the unit disk 𝕌, where ψ is a convex function such that ψ(0) = 1, ψ′(0) > 0, ℜ(ψ(ξ)) > 0 and ψ(𝕌) is symmetric with respect to the real axis. We obtain the sharp solution of Fekete-Szegö problem for the family $\begin{array}{} \mathcal {S}^*_\psi \end{array}$, and then extend the result to the case of corresponding subclass defined on the bounded starlike circular domain Ω in several complex variables, which give an unified answer of Fekete-Szegö problem for the kinds of subclasses of starlike mappings defined on Ω. At last, we propose two conjectures related the same problems on the unit ball in a complex Banach space and on the unit polydisk in ℂn.

Coefficients estimate for a subclass of holomorphic mappings on the unit polydisk in Cn

“…Thus, when γ = 0 in (II) of Theorem 3.3, the result coincide with the main Theorem proved by Xu-Liu-Liu [31]. (b) When γ = 0 in (I) of Theorem 3.3, the result is the conjecture proposed by Tu-Xiong [25]. (c) By choosing suitable functions ψ and real numbers γ as Remark 1.2, the solutions on Fekete-Szegö problems for kinds of subclasses of starlike mappings on U n can be deduced by our main Theorems immediately.…”

“…This is a continuation of the works in [25] and [31]. Our results extend some works that are related starlike mappings in C n , and give a positive answer to a conjecture proposed by Tu-Xiong [25]. Compare with the recent works on Fekete-Szeg ö problem(e.g., [12], [31]), the critical processes of proofs are different: our arguments in this paper are heavily based on the subordination techniques.…”

“…In this paper, we will obtain the sharp coefficients bounds on Fekete-Szeg ö problem for the class S * ψ,γ (U n ). This is a continuation of the works in [25] and [31]. Our results extend some works that are related starlike mappings in C n , and give a positive answer to a conjecture proposed by Tu-Xiong [25].…”

“…Some previous results are improved. Also, the main works give a positive answer to a conjecture proposed by Tu-Xiong [25].…”

“…Recently, Liu-Xu [13] established inequalities between the second and the third coefficients of homogeneous expansions for starlike mappings and starlike mappings of order α defined on bounded starlike circular domains in C n , respectively. Some more general works on coefficients inequalities in several complex variables can be found in Tu-Xiong [25] and Xu-Liu-Liu [31].…”