On the absolute length of polynomials having all zeros in a sector

Let a be an algebraic integer whose all conjugates lie in a sector vertical bar arg z vertical bar <= theta, 0 <= theta <= 90 degrees. Using the method of auxiliary functions, we first improve the known lower bounds of the absolute length of totally positive algebraic integers, i.e., when theta is equal to 0. Then, for 0 < theta < 90 degrees, we compute the greatest lower bound c(theta) of the absolute length of a, for theta belonging to eight subintervals of [0, 90 degrees). Moreover, we have a complete subinterval, i.e., an interval on which the function c(theta) describing the minimum on the sector vertical bar arg(z)vertical bar <= theta is constant, with jump discontinuities at each end. Finally, we obtain an upper bound for the integer transfinite diameter of the interval [0, 1] from the lower bound of the absolute length. The polynomials involved in the auxiliary functions are found by our recursive algorithm. (C) 2014 Elsevier Inc. All rights reserved.