arXiv subjects
Publications and source records attributed to A. O. Kuryliak.
For the Dirichlet series of the form $\displaystyle F(z,ω)=\sum\nolimits_{k=0}^{+\infty} f_k(ω)e^{zλ_k(ω)} $ $ (z\in\mathbb{C},$ $ω\inΩ)$ with pairwise independent real exponents $(λ_k(ω))$ on probability space $(Ω,\mathcal{A},P)$ an estimates of abscissas convergence and absolutely convergence are established.
In this paper we prove some analogue of Wiman's type inequality for random analytic functions in the polydisc $\mathbb{D}^p=\{z\in\mathbb{C}^p\colon |z_j|<1, j\in\{1,\ldots,p\}\},\ p\in\mathbb{Z}_+$. The obtained inequality is sharp.
In this paper we consider a random entire function of the form $f(z,ω)=\sum\nolimits_{n=0}^{+\infty}ξ_n(ω)a_nz^n,$ where $ξ_n(ω)$ are independent standard\break complex gaussian random variables and $a_n\in\mathbb{C}$ satisfy the relations\break $\varlimsup\limits_{n\to+\infty}\sqrt[n]{|a_n|}=0$ and $ \#\{n\colon a_n\neq0\}=+\infty.$ We investigate asymptotic properties of the probability $P_0(r)=P\{ω\colon f(z,ω)$ has no zeros inside $r\mathbb{D}\}.$ Denote $ p_0(r)=\ln^-P_0(r),\ N(r)=\#\{n\colon \ln (|a_n|r^n)>0\},$ $ s(r)=\sum_{n=0}^{+\infty}\ln^+(|a_n|r^{n}). $ Assuming that $a_0\neq0$ we prove that $ 0\leq\varliminf_{r\to+\infty,\ r\notin E}\frac{\ln(p_0(r)- s(r))}{\ln s(r)},\ \varlimsup_{r\to+\infty,\ r\notin E}\frac{\ln(p_0(r)- s(r))}{\ln s(r)}\leq\frac12, $ $ \lim\limits_{r\to+\infty,\ r\notin E}\frac{\ln(p_0(r)- s(r))}{\ln N(r)}=1. $ where $E$ is a set of finite logarithmic measure. Remark that the previous inequalities are sharp. Also we give an answer to open question from \cite[p. 119]{nishry 5}.
For entire functions $f(z)=\sum_{n=0}^{+\infty}a_nz^n, z\in {\Bbb C},$ P. L${\rm \acute{e}}$vy (1929) established that in the classical Wiman's inequality $M_f(r)\leqμ_f(r)\times $ $\times(\lnμ_f(r))^{1/2+\varepsilon},\ \varepsilon>0,$ which holds outside a set of finite logarithmic measure, the constant 1/2 can be replaced almost surely in some sense, by 1/4; here $M_f(r)=\max\{|f(z)|\colon |z|=r\},\ μ_f(r)=\max\{|a_n|r^n\colon n\geq0\},\ r>0. $ In this paper we prove that the phenomenon discovered by P. L${\rm\acute{e}}$vy holds also in the case of Wiman's inequality for entire functions of several variables, which gives an affirmative answer to the question of A. A. Goldberg and M. M. Sheremeta (1996) on the possibility of this phenomenon.
Let $f(z)=\sum_{n=0}^{+\infty} a_nz^n$\ $(z\in\mathbb{C})$\ be an analytic function in the unit disk and $f_t$ be an analytic function of the form $f_t(z)=\sum_{n=0}^{+\infty} a_ne^{iθ_nt}z^n,$ where $t\in\mathbb{R},$ $θ_n\in\mathbb{N},$ and $h$ be a positive continuous function on $(0, 1)$ increasing to $+\infty$ and such that $ \int_{r_0}^1h(r)dr=+\infty, r_0\in(0,1).$\ If the sequence $(θ_n)_{n\geq0}$ satisfies the inequality $$ \varlimsup_{n\to+\infty}\frac1{\ln n}\ln\frac{θ_n}{θ_{n+1}-θ_n}\leqδ\in[0,1/2), $$ then for all analytic functions $f_t$ almost surely for $t$ there exists a set $E=E(δ,t)\subset(0,1)$ such that $\int_Eh(r)dr<+\infty$ and $$ \varlimsup_{{substack} {r\to1-0 r\notin E}{substack}} \frac{\ln M_f(r,t)-\lnμ_f(r)}{2\ln h(r)+\ln\ln\{h(r)μ_f(r)\}}\leq\frac{1+2δ}{4+3δ}, $$ where $M_f(r,t)=\max\{|f_t(z)|\colon |z|=r\},$\ $μ_f(r)=\max\{|a_n|r^n\colon n\geq 0\}$\ for $r\in[0, 1).$