The growth of Weierstrass canonical products of genus zero with random zeros - Zakharko, Yu.B.; Filevych, P.V. (Carpathian Mathematical Publications, 2013)
Let ζ=(ζ_n) be a complex sequence of genus zero, τ be its exponent of convergence, N(r) be its integrated counting function, π(z)=(1-zζ_n) be the Weierstrass canonical product, and M(r) be the maximum modulus of this product. Then, as is known, the Wahlund-Valiron inequality _r→+∞N(r)M(r)w(τ),w(τ):=πτπτ, holds, and this inequality is sharp. It is proved that for the majority (in the probability sense) of sequences ζ the constant w(τ) can be replaced by the constant w(τ2) in the Wahlund-Valiron inequality.