Related papers: Value Distribution and Picard-type Theorems for To…
The classic Schneider-Lang theorem in transcendence theory asserts that there are only finitely many points at which algebraically independent complex meromorphic functions of finite order of growth can simultaneously take values in a…
We construct a certain completion $C^\infty_\mathfrak{g}$ of the universal enveloping algebra of a triangular real Lie algebra $\mathfrak{g}$. It is a Fr\'echet-Arens-Michael algebra that consists of elements of polynomial growth and…
In this paper, we prove some normality criteria concerning transitivity of normality from one family of meromorphic functions to another which improve and generalize some recent results. We also prove some value distribution results for…
In the space $\mathbb{C}$ of the parameters $\lambda$ of the unicritical polynomials family $f(\lambda,z)=f_\lambda(z)=z^d+\lambda$ of degree $d>1$, we establish a quantitative equidistribution result towards the bifurcation current (indeed…
We consider the classical Picard's problem for non-parabolic complete K\"ahler manifolds with non-negative Ricci curvature. Based on the global Green function approach, we give a positive answer to Picard's problem under certain condition…
Let $f$ be a nonzero holomorphic function in the unit ball $\mathbb B$ of the $n$-dimensional complex Euclidean space $\mathbb C^n$ such that the function $f$ vanishes on the set ${\sf Z}\subset \mathbb B$ and satisfies the constraint…
We show that if a meromorphic function has a direct singularity over infinity, then the escaping set has an unbounded component and the intersection of the escaping set with the Julia set contains continua. This intersection has an…
Let f be a non constant meromorphic function and a(not identically zero or infinity) be a meromorphic function satisfying T(r,a) = o(T(r,f)) as r tends to infinity, and p(z) be a polynomial of degree n greater than or equal to 1 with p(0) =…
The main goal of this paper is to establish the higher-dimensional Nevanlinna theory in tropical geometry. We first develop a theory of tropical meromorphic functions ( holomorphic maps) in several variables, such as the proximity function,…
The definition of a non-trivial space of generalized functions of a complex variable allowing to consider derivatives of continuous functions is a non-obvious task, e.g. because of Morera theorem, because distributional Cauchy-Riemann…
This paper establishes a version of Nevanlinna theory based on Jackson difference operator $D_{q}f(z)=\frac{f(qz)-f(z)}{qz-z}$ for meromorphic functions of zero order in the complex plane $\mathbb{C}$. We give the logarithmic difference…
In this paper, we study the value distributions of perfect nonlinear functions, i.e., we investigate the sizes of image and preimage sets. Using purely combinatorial tools, we develop a framework that deals with perfect nonlinear functions…
In this paper, we study the uniqueness of the difference of meromorphic functions. We prove the following result: Let $f$ be a non-constant meromorphic function of hyper-order less than $1$, let $\eta$ be a non-zero complex number,…
We describe some results of value distribution theory of holomorphic curves and quasiregular maps, which are obtained using potential theory. Among the results discussed are: extensions of Picard's theorems to quasiregular maps between…
In this paper, a normality criterion concerning a sequence of meromorphic functions and their differential polynomials is obtained. Precisely, we have proved: Let $\left\{f_j\right\}$ be a sequence of meromorphic functions in the open unit…
In this paper, we study uniqueness problems for entire functions that partially share two values with their higher-order derivatives. The results obtained here both improve and generalize the related results of Li and Yi \cite{LYi}, L\"{u}…
We consider transcendental meromorphic functions for which the zeros, 1-points and poles are distributed on three distinct rays. We show that such functions exist if and only if the rays are equally spaced. We also obtain a normal family…
In this paper we aim to establish some results depending on the comparative growth properties of composite transcendental entire or meromorphic functions and some special type of differential polynomials generated by one of the factors on…
On a complete Riemannian manifold $(M,g)$, we consider $L^{p}_{loc}$ distributional solutions of the the differential inequality $-\Delta u + \lambda u \geq 0$ with $\lambda >0$ a locally bounded function that may decay to $0$ at infinity.…
We show that for a real transcendental meromorphic function f, the differential polynomial f'+f^m with m > 4 has infinitely many non-real zeros. Similar results are obtained for differential polynomials f'f^m-1. We specially investigate the…