论文标题
关于最大模量和随机整个功能的最大模量和零的不平等
Inequalities Concerning Maximum Modulus and Zeros of Random Entire Functions
论文作者
论文摘要
令$f_Ω（z）= \ sum \ limits_ {j = 0}^{\ infty}χ_j（ω）a_j z^j $是一个随机的整个函数，其中$χ_j（ω）$是独立的，并且在可能性空间$（ω，\ nathcalcalcalcal calcal {f} $）上定义了$χ_j（ω）$。在本文中，我们首先定义了一个随机整个功能的家族，其中包括高斯，rademacher，Steinhaus整个功能。然后，我们证明，对于家庭中几乎所有功能，对于任何常数$ c> 1 $，存在一个常数$ r_0 = r_0（ω）$和一个有限对数的$ e \ subset [e，\ infty）$的$ r> r_0 $和$ r_0 $和$ r \ r \ r \ notin e $，$，$，f _ f _ f _ f _ f _ f y | f）， （c/a）^{\ frac1 {b}} \ log^{\ frac1 {b}} \ log m（r，f） +\ log \ log \ log \ log m（r，f），\ qquad a.s. $$，其中$ a，b $是常数，$ m（r，f）$是最大模量，$ n（r，0，f）$是$ f $的加权计数 - 零函数。作为我们主要结果的副产品，我们证明了Nevanlinna的第二个主要定理，用于随机整个功能。因此，家庭中几乎所有功能的特征功能都由称重的计数函数界定，而不是经典的nevanlinna理论中的两个加权计数函数。例如，我们表明，对于几乎所有高斯全部功能$f_Ω$，对于任何$ε> 0 $，有$ r_0 $，因此，对于$ r_0 $，$ r_0 $，$$ t（r，f）\ le n（r，0，f_Ω）+（\ frac12+hog log t（r，r，f）。 $$
Let $f_ω(z)=\sum\limits_{j=0}^{\infty}χ_j(ω) a_j z^j$ be a random entire function, where $χ_j(ω)$ are independent and identically distributed random variables defined on a probability space $(Ω, \mathcal{F}, μ)$. In this paper, we first define a family of random entire functions, which includes Gaussian, Rademacher, Steinhaus entire functions. Then, we prove that, for almost all functions in the family and for any constant $C>1$, there exist a constant $r_0=r_0(ω)$ and a set $E\subset [e, \infty)$ of finite logarithmic measure such that, for $r>r_0$ and $r\notin E$, $$ |\log M(r, f)- N(r,0, f_ω)|\le (C/A)^{\frac1{B}}\log^{\frac1{B}}\log M(r,f) +\log\log M(r, f), \qquad a.s. $$ where $A, B$ are constants, $M(r, f)$ is the maximum modulus, and $N(r, 0, f)$ is the weighted counting-zero function of $f$. As a by-product of our main results, we prove Nevanlinna's second main theorem for random entire functions. Thus, the characteristic function of almost all functions in the family is bounded above by a weighed counting function, rather than by two weighted counting functions in the classical Nevanlinna theory. For instance, we show that, for almost all Gaussian entire functions $f_ω$ and for any $ε>0$, there is $r_0$ such that, for $r>r_0$, $$ T(r, f) \le N(r,0, f_ω)+(\frac12+ε) \log T(r, f). $$
