复变函数
In this paper we investigate bounds for the zeros of a bicomplex polynomial using matrix method. In particular, we find analogue of Gershgorin disk theorem, Cauchy Theorem, theorem of Fujiwara, Walsh and other theorems concerning to zeros…
We consider the connection of functional decompositions of rational functions over the real and complex numbers, and a question about curves on a Riemann sphere which are invariant under a rational function.
In this paper, we attempt to resolve the singularities of the zero variety of a $C^{\infty}$ function of two variables as much as possible by using ordinary blowings up. As a result, we formulate an algorithm to locally express the zero…
Given a bounded planar domain $D$ with $W^{k+1, \infty}$ boundary, $ k\in \mathbb Z^+$, and a weight $\mu\in A_p, 1<p<\infty$, we show that the corresponding truncated Beurling transform is a bounded operator sending $W^{k, p}(D, \mu)$ into…
We study two minimization questions: the nature of curves $\gamma \subset \mathbb{H}$ which minimize the Loewner energy among all curves from 0 to a fixed $z_0 \in \mathbb{H}$, and the nature of $\gamma$ which minimize the Loewner energy…
We study foliations by chord-arc Jordan curves of the twice punctured Riemann sphere $\mathbb C \smallsetminus \{0\}$ using the Loewner-Kufarev equation. We associate to such a foliation a function on the plane that describes the "local…
Let $ \mathcal{H}(\Omega) $ be the class of complex-valued functions harmonic in $ \Omega\subset\mathbb{C} $ and each $f=h+\overline{g}\in \mathcal{H}(\Omega)$, where $ h $ and $ g $ are analytic. In the study of Bohr phenomenon for certain…
The paper continues the author's research in the problem of quantitative investigation of basic curvelinear quasiinvariants of quasiconformal curves. It concerns polygons with infinite number of vertices and provides various distortion…
Let $X$ be a compact Riemann surface and $\mathcal L$ be a positive line bundle on it. We study the conditional zero expectation of all the holomorphic sections of $\mathcal L^n$ which do not vanish on $D$ for some fixed open subset $D$ of…
We investigate the problem of holomorphic algebraizibility for real hypersurfaces in complex space. We introduce a new invariant of a (real-analytic) Levi-nondegenerate hypersurface called {\em the jet transcendence degree}. Using this…
In this paper, we establish various sufficient conditions for a family of holomorphic mappings on a domain $D\subseteq\mathbb{C}$ into $\mathbb{P}^n$ to be normal. Our results are improvements to the Montel-Carath\'eodory Theorem for a…
In this article, we prove a normality criterion for a family of meromorphic functions having zeros with some multiplicity which involves sharing of a holomorphic function by the members of the family. Our result generalizes Montel's…
Schwick, in [6], states that let $\mathcal{F}$ be a family of meromorphic functions on a domain $D$ and if for each $f\in\mathcal{F}$, $(f^n)^{(k)}\neq 1$, for $z\in D$, where $n, k$ are positive integers such that $n\geq k+3$, then…
In this paper, we present various sufficient conditions for a family of meromorphic mappings on a domain $D\subset \mathbb{C}^m$ into $\mathbb{P}^n$ to be meromorphically normal. Meromorphic normality is a notion of sequential compactness…
In this note we regard non-commutative probability theory with operator-valued expectation. We show that the moment generating functions of distributions coming from monotone increment processes of unitary random variables yield…
In this article, we give a Zalcman type renormalization result for the quasinormality of a family of holomorphic functions on a domain in $\mathbb{C}^n$ that takes values in a complete complex Hermitian manifold.
In this article we prove some normality criteria for a family of meromorphic functions which involves sharing of a non-zero value by certain differential monomials generated by the members of the family. These results generalizes some of…
Let D be a domain, n, k be positive integers and n >= K+3. Let F be a family of functions meromorphic in D. If each f in F satisfies (f^n)^(k) not equal to 1 for z in D, then F is normal family. This result was proved by Schwick. In this…
In this article, we prove a normality criterion for a family of meromorphic functions which involves sharing of holomorphic functions. Our result generalizes some of the results of H. H. Chen, M. L. Fang and M. Han, Y. Gu.
In this article, we prove some normality criteria for a family of meromorphic functions having zeros with some multiplicity. Our main result involves sharing of a holomorphic function by certain differential polynomials. Our results…