复变函数

In Geometric function theory, occasionally attempts have been made to solve a particular problem for the Ma-Minda classes, $\mathcal{S}^*(\psi)$ and $\mathcal{C}(\psi)$ of univalent starlike and convex functions, respectively. Recently, a…

In the present paper, we show that given a compact K\"ahler manifold $(X,\omega)$ with a K\"ahler metric $\omega$, and a complex submanifold $V\subset X$ of positive dimension, if $V$ has a holomorphic retraction structure in $X$, then any…

In this paper we consider generalized Hardy spaces in the octonionic setting associated to arbitrary Lipschitz domains where the unit normal field exists almost everywhere. First we discuss some basic properties and explain structural…

For $1 < r \le 2$, we study the set of monomial convergence for spaces of holomorphic functions over $\ell_r$. For $ H_b(\ell_r)$, the space of entire functions of bounded type in $\ell_r$, we prove that $\mbox{mon} H_b(\ell_r)$ is exactly…

Let $\omega$ and $\nu$ be radial weights on the unit disc of the complex plane such that $\omega$ admits the doubling property $\sup_{0\le r<1}\frac{\int_r^1 \omega(s)\,ds}{\int_{\frac{1+r}{2}}^1 \omega(s)\,ds}<\infty$. Consider the one…

For 0<p<1 and f a function in the Hardy space of order p its primitive belongs to the Hardy space q=p/1-p. We show that generically the primitive does not belong, even not locally, in any Hardy space smaller than the Hardy space of order q.

We provide two new formulas for quasiconformal extension to $\overline{\mathbb{C}}$ for harmonic mappings defined in the unit disk and having sufficiently small Schwarzian derivative. Both are generalizations of the Ahlfors-Weill extension…

We characterize the monodromies of projective structures with fuchsian-type singularities. Namely, any representation from the fundamental group of a Riemann surface of finite-type in $PSL_2(\mathbb{C})$ can be represented as the holonomy…

In this paper, we first study the definition and the continuity of the complex Hessian operator associated to an $m$-positive closed current $T$, for some classes of unbounded $m$-subharmonic functions as well as when we consider a…

In his notebooks, Ramanujan presented without proof many remarkable formulae for the solutions to generalized modular equations. Much later, proofs of the formulae were provided by making use of highly nontrivial identities for theta series…

Let $f$ be a transcendental entire function. It was shown in a previous paper that the holomorphic flow $\dot z = f(z)$ always has infinitely many trajectories tending to infinity in finite time. It will be proved here that such…

A recent result of Griffin, Ono, Rolen and Zagier on Jensen polynomials related with the Riemann zeta function is improved.

We consider those elements of the Schwartz algebra of entire functions which are Fourier-Laplace transforms of invertible distributions with compact supports on the real line. These functions are called invertible in the sense of…

We study those real $\mathcal{C}^\infty$ foliations in complex surfaces whose leaves are holomorphic curves. The main motivation is to try and understand these foliations in neighborhoods of curves: can we expect the space of foliations in…

We study questions of existence and uniqueness of quadrature domains using computational tools from real algebraic geometry. These problems are transformed into questions about the number of solutions to an associated real semi-algebraic…

We present some properties of the Weierstrass $\wp$-function associated to the hexagonal (or triangular) lattice. In particular, with the help of an old theorem of I.N. Baker \cite{B} on the characterization of meromorphic solutions of the…

In 1999 Khabibullin established the best estimate in Paley problem for a plurisubharmonic function $u$ of finite lower order $0\leq\rho\leq 1$. For $\rho>1$ obtaining a sharp estimate has remained an open question. In this work we solve…

A very popular model in machine learning is the feedforward neural network (FFN). The FFN can approximate general functions and mitigate the curse of dimensionality. Here we introduce FFNs which represent sections of holomorphic line…

In this paper, we will establish the inequivalence of closed balls and the closure of open balls under the Carath\'{e}odory metric in some planar domains of finite connectivity greater than $2$, and hence resolve a problem posed by…

We give characterizations of (quasi-)plurisubharmonic functions in terms of $L^p$-estimates of $\bar\partial$ and $L^p$-extensions of holomorphic functions.