Theorems of Tauberian type on the distribution of zeros of holomorphic functions

A. A. Shkalikov

Sbornik. Mathematics, Tome 51 (1985) no. 2, pp. 315-344

Résumé

Let $f(\lambda)$ and $g(\lambda)$ be holomorphic functions of finite order in a sector $\Lambda$, and let $n(f,r)$ and $n(g,r)$ be the distribution functions of their zeros inside this sector. Theorems established in this article permit the assertion that $n(f,r)$ and $n(g,r)$ are equivalent if $f(\lambda)$ and $g(\lambda)$ differ “little” on the boundary of $\Lambda$. In the second part of the article domains bounded by curves of parabola type are considered instead of a sector $\Lambda$, and theorems are established which generalize and strengthen Tauberian theorems with a remainder for the distributions of zeros of entire functions and for Stieltjes transforms. Bibliography: 28 titles.