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- W3112909118 abstract "Let $f_omega(z)=sumlimits_{j=0}^{infty}chi_j(omega) a_j z^j$ be a random entire function, where $chi_j(omega)$ are independent and identically distributed random variables defined on a probability space $(Omega, mathcal{F}, mu)$. 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 $r_0=r_0(omega)$ and a set $Esubset [e, infty)$ of finite logarithmic measure such that, for $r>r_0$ and $rnotin E$, $$ |log M(r, f)- N(r,0, f_omega)|le (C/A)^{frac1{B}}log^{frac1{B}}log M(r,f) +loglog M(r, f), qquad a.s. $$ 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 $f_omega$ and for any $epsilon>0$, there is $r_0$ such that, for $r>r_0$, $$ T(r, f) le N(r,0, f_omega)+(frac12+epsilon) log T(r, f). $$" @default.
- W3112909118 created "2020-12-21" @default.
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- W3112909118 date "2020-12-14" @default.
- W3112909118 modified "2023-09-27" @default.
- W3112909118 title "Inequalities Concerning Maximum Modulus and Zeros of Random Entire Functions" @default.
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