Peter Dovbush

• dr.hab
• Researcher at Academy of Sciences of Moldova

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We study normal holomorphic mappings on complex spaces and complex manifolds. Applications are provided.

In this paper, as an application of Zalcman's lemma in $\mathbb{C}^n$, we give a sufficient condition for normality of holomorphic functions of several complex variables, which generalizes previous known one-dimensional criterion of A.J. Lohwater and Ch. Pommerenke \cite[Theorem 1]{MR0338381}.

The aim of this note is to give a proof of the Schottky theorem in general domains in $\mathbb{C}^n$. The proof is short and works for the cases $n = 1$ and $n > 1$ at the same time.

The aim of this paper is to give some applications of Marty’s Criterion and Zalcman’s Rescalling Lemma.KeywordsMarty’s CriterionZalcman’s lemmaZalcman-Pang’s lemmaNormal familiesHolomorphic functions of several complex variables Mathematics Subject Classification (2010) Primary 32A19

We show that a family F = { f } of functions holomorphic in a domain Ω ⊂ C n is normal if all eigenvalues of the complex Hessian matrix of log ⁡ ( 1 + | f | 2 ) are uniformly bounded away from zero on compact subsets of Ω.

The aim of this paper is to give a proof of Zalcman–Pang's Rescalling Lemma in C n .

In this paper, as an application of Zalcman’s lemma in \(C^n,\) we give a sufficient condition for normality of a family of holomorphic functions of several complex variables, which generalizes previous known one-dimensional results of H.L. Royden and W. Schwick.

The aim of this paper is to give a proof of improving of Zalcman's lemma.

The aim of this paper is to give some applications of Zalcman's Rescalling Lemma.

The aim of this paper is to give a proof of Zalcman's Rescalling Lemma in Cn.

Let D be a bounded domain in ℂn . A holomorphic function f: D → ℂ is called normal function if f satisfies a Lipschitz condition with respect to the Kobayashi metric on D and the spherical metric on the Riemann sphere ̅ℂ. We formulate and prove a few Lindelöf principles in the function theory of several complex variables.

Extension of classical Mandelbrojt's criterion for normality of a family of holomorphic zero-free functions of several complex variables is given. We show that a family of holomorphic functions of several complex variables whose corresponding Levi form are uniformly bounded away from zero is normal.

The aim of the present article is to establish the connection between the existence of the limit along the normal and an admissible limit at a fixed boundary point for holomorphic functions of several complex variables.

In multidimensional case we give an extension of the Lindelöf–Gehring–Lohwater theorem involving two paths. A classical theorem of Lindelöf asserts that if f is a function analytic and bounded in the unit disc U which has the asymptotic value L at a point ξ∂U then it has the angular limit L at ξ. Later Lehto and Virtanen proved that a normal functi...

Let D be a convex bounded domain in a complex Banach space. A holomorphic function f : D → is called a normal function if the family f = {f ○ : (Δ, D)} forms a normal family in the sense of Montel (here (Δ, D) denotes the set of all holomorphic maps from the complex unit disc into D). Let {x n } be a sequence of points in D which tends to a boundar...

We give an extension of the Lindelöf-Lehto-Virtanen theorem for normal functions and the Lindelöf-Gehring-Lohwater theorem involving two paths for bounded functions to the multi-dimensional case.

The purpose of the present article is to give the version of the Lindelöf principle which is valid in bounded domains in with C 2-smooth boundary. We also prove that if a Bloch function is bounded on a K-special curve ending at a given boundary point, it is bounded on any admissible domain with vertex at the same point.

Let X be a complex Banach manifold. A holomorphic function f : X → ℂ is called a normal function if the family Ff = {f oφ : φ ∈ O(Δ, X)} forms a normal family in the sense of Montel (here O(Δ, X) denotes the set of all holomorphic maps from the complex unit disc into X). Characterizations of normal functions are presented. A sufficient condition fo...

Let X be a complex Banach manifold. A holomorphic function f : X ! C is called a Bloch function if the family Ff = ff -'¡f('(0)) : ' : ¢ ! X is holomorphicg, ¢ = fz 2 C : jzj < 1g; is a normal family in the sense of Montel. In this paper Bloch functions on complex Banach manifolds are studied. The main result shows that many of the equivalent defln...

Let D be a complete hyperbolic domain in n, n >1, and N a compact Hermitian manifold. We prove a criterion for the existence of the K-limit of an arbitrary holomorphic mapf: D N at an arbitrary boundary point D under the condition that f has the corresponding radial limit at this point.

This is a survey of achievements in the theory of normal holomorphic mappings. We systematize and present all the results on the subject that are obtained by the author from the beginning of the theory until the date of writing the paper. Mathematics subject classification: 32A18.

The “radial” polynomiality criterion for entire functions of several complex variables is proved.

The existence of admissible limits (in Fatou's theorem in the space C ~ , n > i) was discovered by Koranyi [i] and Stein [2]; the complex geometrical nature of this phenomenon has been investigated by Chirka [3]. The aim of the present article is to establish the connection between the existence of the limit along the normal and an admissible limit...

The sufficient conditions for existence almost everywhere of the admissible limit for normal holomorphic functions of several complex variables are given.

The aim of this note is to obtain a sufficient condition for a normal function of several complex variables to be identically equal to zero.

The aim of this note is to obtain a criterion for entire functions of several complex variables to be polynomials.