Fritz Herzog

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

Completely Tetrahedral Sextuples

(1959). Completely Tetrahedral Sextuples. The American Mathematical Monthly: Vol. 66, No. 6, pp. 460-464.

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

Semi-regular plane polygons of integral type

A polygon is called semi-regular if its interior angles are equal to one another. The paper deals mainly with semi-regular polygons whose sides have integral lengths.