Loewner chains and parametric representation of biholomorphic mappings in complex Banach spaces

Let X be a complex Banach space and let B be the unit ball of X. In this paper we obtain sufficient conditions for biholomorphic mappings on B to have parametric representation. Also we study certain properties of Loewner chains, and we obtain infinite dimensional versions of some well known univalence criteria on the unit ball of

Convex mappings in several complex variables

Let B be the unit ball of Cn with respect to an arbitrary norm. We will give a sufficient condition for a local diffeomorphism of C1 class on B to be univalent and to have a convex image. Finally, we present an application on the complex ellipsoid B(p1, ... , pn), where p1, ... , pn ≥ 1

Characterizations of composition operators on Bloch and Hardy type spaces

The main purpose of this paper is to investigate characterizations of composition operators on Bloch and Hardy type spaces. Initially, we use general doubling weights to study the composition operators from harmonic Bloch type spaces on the unit disc $\mathbb{D}$ to pluriharmonic Hardy spaces on the Euclidean unit ball $\mathbb{B}^n$. Furthermore, we develop some new methods to study the composition operators from harmonic Bloch type spaces on $\mathbb{D}$ to pluriharmonic Bloch type spaces on $\mathbb{D}$. Additionally, some application to new characterizations of the composition operators between pluriharmonic Lipschitz type spaces to be bounded or compact will be presented. The obtained results of this paper provide the improvements and extensions of the corresponding known results.Comment: 22page

Starlike mappings of order alpha on the unit ball in complex Banach spaces

In this paper, we will give the growth theorem of starlike mappings of order α on the unit ball B in complex Banach spaces. We also give an analytic sufficient condition for a locally biholomorphic mapping on B to be a starlike mapping of order α

Loewner PDE in infinite dimensions

In this paper, we prove the existence and uniqueness of the solution $f(z,t)$ of the Loewner PDE with normalization $Df(0,t)=e^{tA}$, where $A\in L(X,X)$ is such that $k_+(A)<2m(A)$, on the unit ball of a separable reflexive complex Banach space $X$. We also give improvements of the results obtained recently by Hamada and Kohr, but we omit their proofs for the sake of brevity. In particular, we obtain the biholomorphicity of the univalent Schwarz mappings $v(z,s,t)$ with normalization $Dv(0,s,t)=e^{-(t-s)A}$ for $t\geq s\geq 0$, where $m(A)>0$, which satisfy the semigroup property on the unit ball of a complex Banach space $X$. We further obtain the biholomorphicity of $A$-normalized univalent subordination chains under some normality condition on the unit ball of a reflexive complex Banach space $X$. We prove the existence of the biholomorphic solutions $f(z,t)$ of the Loewner PDE with normalization $Df(0,t)=e^{tA}$ on the unit ball of a separable reflexive complex Banach space $X$. The results obtained in this paper give some positive answers to the open problems and conjectures proposed by the authors in 2013

On the complete Kahlerity of complex spaces

We prove that on every reduced Stein space there exists a complete Kahler metric with a globally defined real-analytic potential function. We also give two results which are generalizations of a theorem of Yasuoka on the complete Kahler exhaustion