arXiv subjects
Publications and source records attributed to O. B. Skaskiv.
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, there are obtained growth estimates of entire in $\mathbb{C}^n$ function of bounded $\mathbf{L}$-index in joint variables. They describe the behaviour of maximum modulus of entire function on a skeleton in a polydisc by behaviour of the function $\mathbf{L}(z)=(l_1(z),\ldots,l_n(z)),$ where for every $j\in\{1,\ldots, n\}$ \ $l_j:\mathbb{C}^n\to \mathbb{R}_+$ is a continuous function. We generalised known results of W. K. Hayman, M. M. Sheremeta, A. D. Kuzyk, M. T. Borduyak, T. O. Banakh and V. O. Kushnir for a wider class of functions $\mathbf{L}.$ One of our estimates is sharper even for entire in $\mathbb{C}$ functions of bounded $l$-index than Sheremeta's estimate.
A concept of boundedness of L-index in joint variables (see in Bordulyak M.T. The space of entire in $\mathbb{C}^n$ functions of bounded L-index, Mat. Stud., 4 (1995), 53--58. (in Ukrainian)) is generalised for analytic in a bidisc function. We proved criteria of boundedness of $\mathbf{L}$-index in joint variables which describe local behaviour of partial derivative and give an estimate of maximum modulus on a skeleton of polydisc. Some improvements of known sufficient conditions of boundednees of $\mathbf{L}$-index in joint variables are obtained.
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.
Let $F$ be an entire function represented by absolutely convergent for all $z\in\mathbb{C}$ Dirichlet series of the form $ F(z) = \sum\nolimits_{n=0}^{+\infty} a_{n}e^{zλ_{n}},$\ where a sequence $(λ_n)$ such that $λ_n\in\mathbb{R}\ \ (n\geq0)$, $λ_n\not=λ_k$ for any $n\not=k$ and $(\forall n\geq 0):\ 0\leqλ_n<β:=\sup\{λ_j:\ j\geq0\}\leq +\infty.$ {Let $h$ be non-decrease positive continuous function on $[0,+\infty)$ and $Φ$ increase positive continuous on $[0,+\infty)$ function.} In this paper we {find} the condition {on} $(μ_n)$ and $(λ_n)$ {such that} the relation $F(x+iy)=(1+o(1))a_{ν(x, F)}e^{(x+iy)λ_{ν(x, F)}} $ holds as $x\to +\infty$\ outside some set $E$ of finite logarithmic $h$-measure uniformly in $y\in\mathbb{R}$.
For entire Dirichlet series of the form $F(z)=\sum\limits_{n=0}^{+\infty} a_{n}e^{zλ_n},\ 0\leλ_n\uparrow+\infty\ (n\to+\infty)$, we establish conditions under which the relation $$ F(x+iy)=(1+o(1))a_{ν(x,F)}e^{(x+iy)λ_{ν(x,F)}} $$ is true as $x\to+\infty$ outside some set $E$ such that $\text{ h-meas }(E)=\int_{E}dh(x)<+\infty$ uniformly in $y\in\Bbb{R}$, where $h(x)$ is positive continuous function increasing to $+\infty$ on $[0,+\infty)$ with non-decreasing to $+\infty$ derivative.
We propose a generalisation of analytic in a domain function of bounded index, which was introduced by J. G. Krishna and S. M. Shah \cite{krishna}. In fact, analytic in the unit ball function of bounded index by Krishna and Shah is an entire function. Our approach allows us to explore properties of analytic in the unit ball functions. We proved the necessary and sufficient conditions of bounded $L$-index in direction for analytic functions. As a result, they are applied to study partial differential equations and get sufficient conditions of bounded $L$-index in direction for analytic solutions. Finally, we estimated growth for these functions.
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 $r\in[0, 1).$