Distribution of zeros of sequences of polynomials of unbounded degree

Abstract

With any infinite sequence of polynomials {fn(z) }, where fn(z) is of degree n, and no two polynomials have the same degree, we associate an open set R of the complex z plane defined in terms of the behavior of the moduli jfn(z) I for large n. Precise definitions are given in the sequel. A connected component Ro of R is assumed to contain a point q, which we take to be the origin, such that Jf,,(q) f, when compared asymptotically to sup If,(z) J in Ro, is not too small. (See (13) and (14)). Theorem 1 then shows that the proportion of zeros of fn(z) in any neighborhood of any finite boundary point of Ro is bounded away from zero for some subsequence of the given sequence. Moreover, the number of zeros of fn(z) in any bounded, closed subset of Ro is o(n) for the whole sequence. These results are a generalization of an unpublished theorem of the author's thesis. Ro is called a fiat region of the sequence. Theorems 2 and 3 consider the geometry of Ro. Theorem 2 shows that Ro cannot contain an infinite, open sector if it has a finite boundary point. The immediate interest of Theorem 2 is contained in Example (6), where sectorwise distribution of zeros is proved for a sequence of partial sums of a Taylor's series with one big coefficient occurring anywhere except in the earliest terms. In the corresponding theorem of Erdos and Turan [4; 5 ], while a stronger type of distribution is proved, the last coefficient of each partial sum must be large. Similarly in Dvoretsky [2; 3] the last or next to last coefficient must be large. Theorem 3 shows that any of a wide class of regions is a flat region for some sequence. Theorem 4 is the weak analogue of Theorem 1 when fn(z) is analytic and not necessarily a polynomial. Other examples are given discussing the converse of Theorem 1 and modifications in its hypothesis and conclusions. Apart from one application each of Jensen's Theorem and the Fundamental Theorem of Normal Families, the proofs depend almost entirely on elementary inequalities and simple majorizations. Theorem 1 may be looked upon as a generalization, both in its statement and method of proof, of the well-known theorem of Jentzsch on the angular distribution of zeros of partial sums of Taylor's Series.