Characterizations of Herglotz–Nevanlinna Functions Using Positive Semi-Definite Functions and the Nevanlinna Kernel in Several Variables

In this paper, we give several characterizations of Herglotz–Nevanlinna functions in terms of a specific type of positive semi-definite functions called Poisson-type functions. This allows us to propose a multidimensional analogue of the classical Nevanlinna kernel and a definition of generalized Nevanlinna functions in several variables. Furthermore, a characterization of the symmetric extension of a Herglotz–Nevanlinna function is also given. The subclass of Loewner functions is discussed as well, along with an interpretation of the main result in terms of holomorphic functions on the unit polydisk with non-negative real part.

Linear combinations of composition operators on $H^{\infty }$ spaces over the unit ball and polydisk

In this paper, we characterize completely the compactness of linear combinations of composition operators acting on the space $H^{\infty }(\mathbb{B}_{N})$ H ∞ ( B N ) of bounded holomorphic functions over the unit ball $\mathbb{B}_{N}$ B N from two different aspects. The same problems are also investigated on the space $H^{\infty }(\mathbb{D}^{N})$ H ∞ ( D N ) over the unit polydisk $\mathbb{D}^{N}$ D N .

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

The aim of this paper is to obtain the sharp solutions of Fekete-Szeg? problems of high dimensional version for family of holomorphic mappings that are normalized on the unit polydisk Un in Cn. The main results unify some recent works, which are closely related to the starlike mappings. Moreover, some previous results are improved.

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

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.

SHARP BOUNDS OF SOME COEFFICIENT FUNCTIONALS OVER THE CLASS OF FUNCTIONS CONVEX IN THE DIRECTION OF THE IMAGINARY AXIS

We apply the Schwarz lemma to find general formulas for the third coefficient of Carathéodory functions dependent on a parameter in the closed unit polydisk. Next we find sharp estimates of the Hankel determinant $H_{2,2}$ and Zalcman functional $J_{2,3}$ over the class ${\mathcal{C}}{\mathcal{V}}$ of analytic functions $f$ normalised such that $\text{Re}\{(1-z^{2})f^{\prime }(z)\}>0$ for $z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$, that is, the subclass of the class of functions convex in the direction of the imaginary axis.