Mathematics

In this paper, we will give suitable conditions on differential polynomials Q(f) such that they take every finite non-zero value infinitely often, where f is a meromorphic function in complex plane. These results are related to Problem 1.19 and Problem 1.20 in a book of Hayman and Lingham \cite{HL}. As consequences, we give a new proof of the Hayman conjecture. Moreover, our results allow differential polynomials Q(f) to have some terms of any degree of f and also the hypothesis n>k in \cite[Theorem 2]{BE} is replaced by n≥2 in our result.

We prove the existence of a roof function for arclength null quadrature domains having finitely many boundary components. This bridges a gap toward classification of arclength null quadrature domains by removing an a priori assumption from previous classification results.

Value distribution and uniqueness problems of difference operator of an entire function have been investigated in this article. This research shows that a finite ordered entire function f when sharing a set S={α(z),β(z)} of two entire functions α and β with max{ρ(α),ρ(β)}<ρ(f) with its difference L n c (f)= ∑ n j=0 a j f(z+jc) , then L n c (f)≡f , and more importantly certain form of the function f has been found. The results in this paper improve those given by \emph{k. Liu}, \emph{X. M. Li}, \emph{J. Qi, Y. Wang and Y. Gu} etc. Some constructive examples have been exhibited to show the condition max{ρ(α),ρ(β)}<ρ(f) is sharp in our main result. Examples have been also exhibited to show that if CM sharing is replaced by IM sharing, then conclusion of the main results ceases to hold.

The purpose of this article is to investigate a hyperbolic complex manifold M exhausted by a pseudoconvex domain Ω in C n via an exhausting sequence { f j :Ω→M} such that f −1 j (a) converges to a boundary point ξ 0 ∈∂Ω for some point a∈M .

The goal of this note is to present some recent results of our research concerning multiplier ideal sheaves on complex spaces and singularities of plurisubharmonic functions. We firstly introduce multiplier ideal sheaves on complex spaces (\emph{not} necessarily normal) via Ohsawa's extension measure, as a special case of which, it turns out to be the so-called Mather-Jacobian multiplier ideals in the algebro-geometric setting. As applications, we obtain a reasonable generalization of (algebraic) adjoint ideal sheaves to the analytic setting and establish some extension theorems on Kähler manifolds from \emph{singular} hypersurfaces. Relying on our multiplier and adjoint ideals, we also give characterizations for several important classes of singularities of pairs associated to plurisubharmonic functions. Moreover, we also investigate the local structure of singularities of log canonical locus of plurisubharmonic functions. Especially, in the three-dimensional case, we show that for any plurisubharmonic function with log canonical singularities, its associated multiplier ideal subscheme is weakly normal, by which we give a complete classification of multiplier ideal subschemes with log canonical singularities.

We prove that any smooth mapping between reduced analytic spaces induces a natural pullback operation on smooth differential forms.

We prove that the tangential Cauchy-Riemann operator has closed range on Levi-pseudoconvex CR manifolds that are embedded in a q-convex complex manifold X . Our result generalizes the known case when X is a Stein manifold.

We give a short proof that the limsup of the p-th root of the modulus of the p-th moment of a sequence of complex numbers is equal to the modulus of the maximum of the sequence.This strengthens known results, and provides an analog to a recent result concerning moments of complex polynomials.

The squeezing problem on C can be stated as follows. Suppose that Ω is a multiply connected domain in the unit disk D containing the origin z=0 . How far can the boundary of Ω be pushed from the origin by an injective holomorphic function f:Ω?�D keeping the origin fixed? In this note, we discuss recent results on this problem obtained by Ng, Tang and Tsai (Math. Anal. 2020) and by Gumenyuk and Roth (arXiv:2011.13734, 2020) and also prove few new results using a method suggested in one of our previous papers (Zapiski Nauchn. Sem. POMI 1993).

The issue had been raised whether the Fredholm series z+ z 2 +...+ z 2 n +... has infinitely many zeroes in the unit disk. We provide an affirmative answer, proving that in fact every complex number occurs as a value infinitely many times.