• Corpus ID: 252531931

Zeros and coefficients

  title={Zeros and coefficients},
  author={Alexandre Eremenko},
Two theorems on the asymptotic distribution of zeros of sequences of analytic functions are proved. First one relates the asymptotic behavior of zeros to the asymptotic behavior of coefficients. Second theorem establishes a relation between the asymptotic behaviors of zeros of a function and zeros of derivative. 




This paper is a detailed exposition of the results announced in Soviet Math. Dokl. 17(1976), 1169-1171 (MR 55 #8351). Bibliography: 22 titles.

Value distribution and potential theory

Results discussed are: extensions of Picard's theorems to quasiregular maps between Riemannian manifolds, a version of the Second Main Theorem of Nevanlinna for curves in projective space and non-linear divisors, description of extremal functions in Nevanlinsna theory and results related to Cartan's 1928 conjecture on holomorphic curves in the unit disc omitting hyperplanes.

Zeros of Sections of Power Series: Deterministic and Random

We present a streamlined proof (and some refinements) of a characterization (due to F. Carlson and G. Bourion, and also to P. Erdős and H. Fried) of the so-called Szegő power series. This

Limits of zeroes of recursively defined polynomials.

Substantial conditions are found, subject to mild nondegeneracy conditions, that a number x be a limit of zeroes of {P(n)} in the sense that there is a sequence of polynomials with P(n)(z(n) = 0, z(n)-->x.

Notions of Convexity

The first two chapters of the book are devoted to convexity in the classical sense, for functions of one and several real variables respectively. This gives a background for the study in the

On dichromatic polynomials

Universality for zeros of random analytic functions

Let 0; 1;::: be independent identically distributed (i.i.d.) ran- dom variables such that E log(1 +j 0j) < 1. We consider random analytic functions of the form Gn(z) = 1 X k=0 kfk;nz k ; n

Functions of Completely Regular Growth

1. Entire functions of completely regular growth of one variable.- 1. Preliminaries.- 2. Regularity of growth, D'-convergence and right distribution of zeros.- 3. Rays of completely regular growth.

Value distribution of sequences of rational functions