39 research outputs found
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
Loewner PDE in infinite dimensions
In this paper, we prove the existence and uniqueness of the solution
of the Loewner PDE with normalization , where is
such that , on the unit ball of a separable reflexive complex
Banach space . 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
with normalization for ,
where , which satisfy the semigroup property on the unit ball of a
complex Banach space . We further obtain the biholomorphicity of
-normalized univalent subordination chains under some normality condition on
the unit ball of a reflexive complex Banach space . We prove the existence
of the biholomorphic solutions of the Loewner PDE with normalization
on the unit ball of a separable reflexive complex Banach space
. The results obtained in this paper give some positive answers to the open
problems and conjectures proposed by the authors in 2013
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
On strongly starlikeness of order α in several complex variables
In this paper we introduce the concept of strongly starlikeness of order α > 0, for holomorphic mappings defined on the unit ball of Cn. We obtain the distorsion and the covering theorems for strongly starlike mappings of order α ∈ (0,1] and we give a connection between strongly starlikeness and spirallikeness in Cn
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
Convex subordination chains and injective mappings in Cn
AbstractIn this paper we continue the work related to convex subordination chains in C and Cn, and prove that if f(z)=z+∑k=2∞Ak(zk) is a holomorphic mapping on the Euclidean unit ball Bn in Cn such that ∑k=2∞k2‖Ak‖⩽1, a:[0,1]→[0,∞) is a function of class C2 on (0,1) and continuous on [0,1], such that a(1)=0, a(t)>0, ta′(t)>−1/2 for t∈(0,1), and if a(⋅) satisfies a differential equation on (0,1), then f(z,t)=a(t2)Df(tz)(tz)+f(tz) is a convex subordination chain over (0,1] and the mapping F(z)=a(‖z‖2)Df(z)(z)+f(z) is injective on Bn. We also present certain coefficient bounds which provide sufficient conditions for univalence, quasiregularity and starlikeness for the chain f(z,t). Finally we give some examples of convex subordination chains over (0,1]
Certain partial differential subordinations on some Reinhardt domains in
We obtain an extension of Jack-Miller-Mocanu's Lemma for holomorphic mappings defined in some Reinhardt domains in . Using this result we consider first and second order partial differential subordinations for holomorphic mappings defined on the Reinhardt domain with p ≥ 1