TY - JOUR
AU - Zakharko, Yu.B.
AU - Filevych, P.V.
PY - 2013/06/20
Y2 - 2024/09/15
TI - The growth of Weierstrass canonical products of genus zero with random zeros
JF - Carpathian Mathematical Publications
JA - Carpathian Math. Publ.
VL - 5
IS - 1
SE - Scientific articles
DO - 10.15330/cmp.5.1.50-58
UR - https://journals.pnu.edu.ua/index.php/cmp/article/view/3650
SP - 50-58
AB - <p>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<br />$$<br />\limsup_{r\to+\infty}\frac{N(r)}{\ln M(r)}\ge w(\tau),\qquad w(\tau):=\frac{\sin\pi\tau}{\pi\tau},<br />$$<br />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-ValironÂ inequality.</p>
ER -