tailieunhanh - Đề tài " Cauchy transforms of point masses: The logarithmic derivative of polynomials "

Obviously, () implies the same estimate for M(Z(QN, P)). It was suggested in [2] that in this case the (1+logN) term could be omitted at the cost of multiplying by a constant. The above suggestion means that in the passage from the sum of moduli to the modulus of the sum in () essential cancellation should take place. As a contribution towards this end the authors showed that any straight line L intersects Z(QN, P) in a set FP of linear measure less than 2eP−1N. Further information about the complement of FP under certain conditions on {zk} is obtained in [1]. Clearly we may assume that N. | Annals of Mathematics Cauchy transforms of point masses The logarithmic derivative ofpolynomials By J. M. Anderson and V. Ya. Eiderman Annals of Mathematics 163 2006 1057 1076 Cauchy transforms of point masses The logarithmic derivative of polynomials By J. M. Anderson and V. Ya. Eiderman 1. Introduction For a polynomial N Qn z JJ z - Zk k 1 of degree N possibly with repeated roots the logarithmic derivative is given by QN z y 1 Q z k i Z - Zk For fixed P 0 we define sets Z Qn P and X Qn P by Í JL Z Qn P z Z e C k 1 -IV Z Z e C V k 1 1 Zk - P z - Zk P Z Clearly Z Qn P c X Qn P . Let D z r denote the disk K z e C z - z r . In 2 it was shown that X Qn P is contained in a set of disks D wj rj with centres Wj and radii rj such that E rj P 1 log N j Research supported in part by the Russian Foundation of Basic Research Grant no. 05-01-01021 and by the Royal Society short term study visit Programme no. 16241. The second author thanks University College London for its kind hospitality during the preparation of this work. The first author was supported by the Leverhulme Trust . . 1058 J. M. ANDERSON AND V. YA. EIDERMAN or as we prefer to state it 2N M X Qn P P 1 logN . Here M denotes 1-dimensional Hausdorff content defined by M A inf Tj where the infimum is taken over all coverings of a bounded set A by disks with radii Tj. The question of the sharpness of the bound in was left open in 2 . We prove - Theorem below - that the estimate for X is essentially best possible. Obviously implies the same estimate for M Z Qn P . It was suggested in 2 that in this case the 1 log N term could be omitted at the cost of multiplying by a constant. The above suggestion means that in the passage from the sum of moduli to the modulus of the sum in essential cancellation should take place. As a contribution towards this end the authors showed that any straight line L intersects Z Qn P in a set Fp of linear measure less than 2eP-1N. Further information about the

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