Some notes on the power sums of complex numbers whose sum is 0

Abstract

1. Let z 1,..., z n be complex numbers. The important method of P. Turán, based on estimating from below the maximum of the modulus of the power sums $$S_v = \sum\limits_{j = 1}^n {z_j^v } \,1 \leqq v \leqq n$$ (1) and of generalized power sums under different normations for the numbers z j , provided unexpected new results on quite different fields of mathematics and many new problems emerge in the further development of the theory. One of them, raised by Turán in his lectures on the topic and in his posthumous book [4], is the estimation of the sums (1) under the following conditions: $$s_1 = 0$$ (2) and either $$|z_j | = 1\,for\,some\,j,\,1 \leqq j \leqq n$$ (3) or $$|z_j | \geqq 1\,(j = 1, \ldots ,n)$$ (4)