Зростання канонічних добутків Вейєрштрасса нульового роду з випадковими нулями
| Autor: | Yu. V. Zakharko, P. V. Filevych |
|---|---|
| Rok vydání: | 2013 |
| Předmět: |
рід
Pure mathematics lcsh:Mathematics Mathematics::Number Theory General Mathematics Entire function добуток вейєрштрасса Zero (complex analysis) Order (ring theory) усереднена лічильна функція Function (mathematics) lcsh:QA1-939 Combinatorics максимум модуля показник збіжності ціла функція Product (mathematics) Genus (mathematics) Pi Exponent порядок Mathematics |
| Zdroj: | Karpatsʹkì Matematičnì Publìkacìï, Vol 5, Iss 1, Pp 50-58 (2013) |
| ISSN: | 2313-0210 2075-9827 |
| Popis: | Let $\zeta=(\zeta_n)$ be a complex sequence of genus zero, $\tau$ be its exponent of convergence, $N(r)$ be its integrated counting function, $\pi(z)=\prod\bigl(1-\frac{z}{\zeta_n}\bigr)$ be the Weierstrass canonical product, and $M(r)$ be the maximum modulus of this product. Then, as is known, the Wahlund-Valiron inequality $$ \limsup_{r\to+\infty}\frac{N(r)}{\ln M(r)}\ge w(\tau),\qquad w(\tau):=\frac{\sin\pi\tau}{\pi\tau}, $$ holds, and this inequality is sharp. It is proved that for the majority (in the probability sense) of sequences $\zeta$ the constant $w(\tau)$ can be replaced by the constant $w\left(\frac{\tau}2\right)$ in the Wahlund-Valiron inequality. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|