复变函数

We consider constellations of disks which are unions of disjoint hyperbolic disks in the unit disk with fixed radii and unfixed centers. We study the problem of maximizing the conformal capacity of a constellation with a fixed number of…

We prove several Sobolev-type inequalities related to the $\bar\partial$-operator on bounded domains in $\mathbb{C}^n$, which can be viewed as a $\bar\partial$-version of the classical Sobolev inequality and its various generalizations, and…

Let $\Omega$ be a smooth bounded pseudoconvex domain in $\mathbb{C}^{n}$. Let $1\leq q_{0}\leq (n-1)$. We show that if $q_{0}$--sums of eigenvalues of the Levi form are comparable, then if the Diederich--Forn\ae ss index of $\Omega$ is $1$,…

In this article, the new inequalities for the weighted sums of coefficients in the class of bounded functions in the disk are obtained. We develop the methods of I.R.~Kayumov and S.~Ponnusamy, using E.~Reich's theorem on the majorization of…

Let $\mathcal{F}$ be a singular Riemann surface foliation on a complex manifold $M$, such that the singular set $E \subset M$ is non-discrete. We study the behavior of the foliation near the singular set $E$, particularly focusing on…

We prove that the Bernardi Integral Operator maps certain classes of bounded starlike functions into the class of convex functions, improving the result of Oros and Oros. We also present a general unified method for investigating various…

Each homeomorphic parametrization of a Jordan curve via the unit circle extends to a homeomorphism of the entire plane. It is a natural question to ask if such a homeomorphism can be chosen so as to have some Sobolev regularity. This…

The primary objective of this work is to obtain some sufficient conditions so that normalized Gauss hypergeometric function satisfies exponential starlikeness and convexity in the unit disk. Moreover, conditions on parameter of this…

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^N$. Given a continuous plurisubharmonic function $u$ on $\Omega$, we construct a sequence of Gaussian analytic functions $f_n$ on $\Omega$ associated with $u$ such that…

In this paper, we characterize the K\"ahler-hyperbolicity length of a bounded symmetric domain, defined by its rank and genus, as a unique constant determined by a constant gradient length of a special Bergman potential. Additionally, we…

We prove that affine maps are uniquely extremal quasiconformal maps on the complement of a well distribute set in the complex plane answering a conjecture from \cite{markovic}. We construct the required Reich sequence using Bergman…

The theory of Khinchin families connects Probability Theory and Complex Analysis. Along this PhD thesis, we exploit this connection to obtain, using a variety of local central limit theorems, asymptotic formulas for the coefficients of…

Let $\mathcal{S}_u^*$ denote the class of all analytic functions $f$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$, normalized by $f(0)=f'(0)-1=0$ that satisfies the inequality $\left|zf'(z)/f(z)-1\right|<1$ in $\mathbb{D}$. In…

Let $\varphi$, $\psi\in C(\mathbb{T})$, $g\in C(\overline{\mathbb{D}})$, where $\mathbb{D}$ and $\mathbb{T}$ denote the unit disk and the unit circle, respectively. Suppose that $f\in C^{4}(\mathbb{D})$ satisfies the following: (1) the…

We construct a quasiregular mapping in $\mathbb{R}^3$ that is the first to illustrate several important dynamical properties: the quasi-Fatou set contains wandering components; these quasi-Fatou components are bounded and hollow; and the…

The main goal of this paper is to obtain sufficient conditions so that Le Roy type functions and multivariate Le Roy type functions satisfy subordination of exponential function. Moreover conditions on parameters have been derived to claim…

We present a unified numerical method to determine the shapes of multiple Hele-Shaw bubbles in steady motion, and in the absence of surface tension, in three planar domains: free space, the upper half-plane, and an infinite channel. Our…

In this note we give a new sufficient condition for the boundedness of the composition operator on the Dirichlet-type space on the disc, via a two dimensional change of variables formula. With the same formula, we characterise the bounded…

In this research notes, we investigate some remain problems in the uniqueness of meromorphic function. Using some deep results of Yamanoii, we obtain some results in this notes.

In this paper, we initiate the study of the geometric function theory for slice starlike functions over quaternions and its subclasses. This allows us to answer negatively some questions about the Bieberbach conjecture, the growth,…