推广的Laguerre不等式及实零点的一个判别法

Laguerre不等式的离散类似和I.克拉西科夫

Discrete analogues of the Laguerre inequalities and a conjecture of I. Krasikov

我的一个猜想克拉西科夫被证明了。几个离散类似的经典多项式不等式的推导，随着结果，允许扩展到一类超越整函数的Laguerre-Pólya类。

Erdélyi多元Laguerre多项式的上界

Upper bounds for Erdélyi's multivariate Laguerre polynomials

本文为Arthur Erdélyi引入的多元Laguerre多项式建立两个不等式，推广了经典的Szegö不等式，并在结论部分给出了上界的比较分析。

Differential inequalities and a Marty-type criterion for quasi-normality

We show that the family of all holomorphic functions $f$ in a domain $D$ satisfying $$\frac{|f^{(k)}|}{1+|f|}(z)\le C \qquad \mbox{ for all } z\in D$$ (where $k$ is a natural number and $C>0$) is quasi-normal. Furthermore, we give a general counterexample to show that for $α>1$ and $k\ge2$ the condition $$\frac{|f^{(k)}|}{1+|f|^α}(z)\le C \qquad \mbox{ for all } z\in D$$ does not imply quasi-normality.

On the entire functions from the Laguerre-Pólya I class having the increasing second quotients of Taylor coefficients

We prove that if $f(x) = \sum_{k=0}^\infty a_k x^k,$ $a_k >0, $ is an entire function such that the sequence $Q := \left( \frac{a_k^2}{a_{k-1}a_{k+1}} \right)_{k=1}^\infty$ is non-decreasing and $\frac{a_1^2}{a_{0}a_{2}} \geq 2\sqrt[3]{2},$ then all but a finite number of zeros of $f$ are real and simple. We also present a criterion in terms of the closest to zero roots for such a function to have only real zeros (in other words, for belonging to the Laguerre--Pólya class of type I) under additional assumption on the sequence $Q.$

正定核的傅里叶变换与Riemann $ξ$ -函数

Fourier transforms of positive definite kernels and the Riemann $ξ$-Function

本文研究可表示为某些容许核的傅里叶变换的整函数零点的分布。主要结果揭示了正定核的Bochner-Khinchin-Mathias理论与广义实Laguerre不等式之间的密切联系。雅可比θ函数的凸性和凸性在整个工作中起着重要的作用。本文最后提出了几个问题和开放的问题。

Szegö kernel asymptotics and Morse inequalities on CR manifolds with $S^1$ action

Let $X$ be a compact connected CR manifold of dimension $2n-1, n\geq 2$. We assume that there is a transversal CR locally free $S^1$ action on $X$. Let $L^k$ be the $k$-th power of a rigid CR line bundle $L$ over $X$. Without any assumption on the Levi-form of $X$, we obtain a scaling upper-bound for the partial Szegő kernel on $(0,q)$-forms with values in $L^k$. After integration, this gives the weak Morse inequalities. By a refined spectral analysis, we also obtain the strong Morse inequalities in CR setting. We apply the strong Morse inequalities to show that the Grauert-Riemenschneider cri…