George Piranian

Metric properties of polynomials

(1) f (2) = iil[ (2 22,) VIZ I (written in the form f(x) = n (x -x,), when only real variables are under consideration). We are concerned primarily with the point set B = E (f) , defined as the set where the inequality 1 f (z)I < 1 is satisfied. In Section 2, we determine the infimum of the length of rhe longest interval in the se1 I? nI+ (L denotes the real axis) for the case where the X, lie on the interval I = I-1 , I] on L. Section 3 deals with the diameter of the set E nL, under the more general hypothesis that the x7, lie on the interval I, = [-I , Y] . In Section 4, we study the two-dimensional measure of the set E, under the restriction that all the 2, lie in the closute D of the unit disk D. We use a theorem of G. R. MacLane to show that the measure 1 B (f)I can be made arbitrarily small. We also deal briefly with the relation between the transfinite diameter of a closed set F and the infimum of II?(f) 1, under the hypothesis that all 2, lie in F. Section 5 is devoted to the problem of finding the greatest number of components thar the set E can have when the .z,) are required to lie in D; Section 4 deals with the sum of the diameters of the components of E, under the hypothesis that all z\. lie in the disk D, : z i c; T. Section 7 concerns polynomials (1) for which the set B is connected. In Section 8, we consider two problems concerning the convexity of E and of the components of E, respectively. And in Section 9, we prove a theorem

STRONG FATOU-1-POINTS OF BLASCHKE PRODUCTS

This paper shows that to every countable set M on the unit circle there corresponds a Blaschke product whose set of strong Fatou-1-points contains M. It also shows that some Blaschke products have an uncountable set of strong Fatou-1 - points.

Level sets of infinite length

The geometry of level sets plays an important role in the analysis of various function-theoretic problems. Often, however, the level sets are so complicated that one must either choose sets associated with special levels (see [3], [4]) or resort to the approximation of level sets by shorter curves (see[l, pp. 550-553]). In [3] and [4], there are some weak estimates on the length e(r, R) of the sets {z: l f (z ) I = R, Izt < r} associated with a function [ meromorphic in the plane. For such an / we do not know whether the quantity ca(r, R) can be bounded in terms of Nevanlinnas characteristic function T without reference to exceptional levels. But the following is implicit in the results in [3, pp. 121-123] and [4, p. 44]: I / [ is meromorphic in lzl-<2r and f (0 )= 1, then each subinterval [a,/3] of (O, oo) contains a set I of measure ([3a)/2 such that the inequality

Laconicity and redundancy of Toeplitz matrices

Abstract : The convergence field of a Toeplitz matrix is a monotonic function of the set of rows that compose the matix, in the sense that the deletion of some of the rows of the matrix (followed by appropriate renumbering of the rows that remain) can never decrease the convergence field. In the case of certain matrices, the deletion of infinitely many rows always increases the convergence field; but there exist matrices that do not have this property. This dichotomy was considered with special reference to the space of bounded sequences and certain classical families of matrices.

The Set of Nondifferentiability of a Continuous Function

(A set is of type GC provided it is the intersection of countably many open sets; without loss of generality, we may assume that the open sets form a decreasing sequence. A set is of type Gb, if it is the union of countably many sets of type GC.) Zahorskis theorem is an obvious consequence of the two lemmas below. We formulate the lemmas in terms of the Dini derivatives D+f, D+f, D-f, D_f. Of these, the upper and lower right-hand derivatives, for example, are defined as

A note on transforms of unbounded sequences

possesses at least one limit point in the finite plane; and it was counter-conjectured that for every regular Toeplitz matrix A there exists a sequence sn such that the sequence f, of equation (1) tends to infinit.y monotonically. It is the purpose of the present note to report that both conjectures are false and to prove a consolation theorem regarding the first conjecture. The notation of equation (1) will be used throughout the paper.

Strongly annular functions with small Taylor coefficients

Daniel D. Bonar 1, Frank Carroll 2, and George Piranian 3 Department of Mathematical Sciences, Denison University, Granville, OH 43023, USA 2 Department of Mathematics, Ohio State University, Columbus, OH 43210, USA a Department of Mathematics, University of Michigan, Ann Arbor, MI 48104, USA

Some properties of the Fejér polynomials

were introduced, by way of their real and imaginary parts on the unit circle C, by Fejér [6; 7]. Fejér showed that the number 24-7T is an upper bound for the modulus both of the real and of the imaginary part of PB on C (w = l, 2, • • • ); a very simple proof that the polynomials Pn are uniformly bounded on C is given in [3, p. 43]. In view of recent applications of the Fejér polynomials in the study of Taylor series (see, for example, [3; 4; 8]), we have undertaken an investigation of their least upper bound on C (see §3) and of the distribution of their zeros (see §2). Elementary considerations show that limn<0O nPn(z) =1/(1 —z) for \z\ <1, and that the convergence is uniform in every disc \z\ gr<l. From this and the fact that the reciprocal of every zero of P„ is also a zero of P„, it follows that the zeros of P„ lie on or near the circle C. The theorem of Jentzsch and Szegö [9; 11] implies further that the arguments of the zeros are uniformly distributed in the interval [0, 27t]. Somewhat stronger results on the distribution of the arguments could be obtained by applying a theorem of Erdös and Turan [5, p. 106]; but by using a method which involves nothing deeper than Rouchés theorem, we prove that each of the « — 1 sectors

On the convergence of certain partial sums of a Taylor series with gaps

M{r) = 1 (r g 0), &n = Xn-|-i/Xw 1. Ostrowski has proved that if {dni} is a sequence extracted from the sequence {dn} such that lim inf 0Wt->0, then every regular point of ƒ(z) on the circle | z\ = 1 is the center of a circle in which the sequence {Sni(z)} converges uniformly to f(z). Restricting ourselves to the question of convergence at the regular points themselves, we shall prove the following theorem :

The zero-sets of the radial-limit functions of inner functions

A set E on the unit circle is the zero-set of the radial-limit function of some inner function if and only if E is a countable intersection of Fa-sets of measure 0.