The growth of Weierstrass canonical products of genus zero with random zeros

  • Yu.B. Zakharko Lviv National University of Veterinary Medicine and Biotechnologies, 50 Pekarska str., 79010, Lviv, Ukraine
  • P.V. Filevych Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine
Keywords: entire function, Weierstrass products, maximum modulus, order, genus, exponent of convergence, integrated counting function
Published online: 2013-06-20

Abstract


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.

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How to Cite
(1)
Zakharko Y., Filevych P. The Growth of Weierstrass Canonical Products of Genus Zero With Random Zeros. Carpathian Math. Publ. 2013, 5 (1), 50-58.