|
VIDEO LIBRARY |
International Conference on Complex Analysis Dedicated to the Memory of Andrei Gonchar and Anatoliy Vitushkin
|
|||
|
On Turán type inequalities related to metric estimates of simple partial fractions M. A. Komarov Vladimir State University |
|||
Abstract: In 1939 P. Turán established the following inequality for the derivative $$ \|P_n'\|_{C[-1,1]}>\frac{\sqrt{n}}{6}\,\|P_n\|_{C[-1,1]}, $$ converse to the classical A. A. Markov inequality (which is true for arbitrary polynomials). In the talk, we discuss a number of generalizations of Turán's inequality, in particular, the case when the zeros of a polynomial are taken on a set larger than the unit interval. The technique of constructing these generalizations uses the apparatus of metric estimates of simple partial fractions (that is, the logarithmic derivatives $\rho_n(x)=P_n'(x)/P_n(x)=\sum_{k=1}^n (x-z_k)^{-1}$ of polynomials $$ \{x\in E: \|\rho_n(x)|\ge \delta\}, \qquad \delta>0 $$ (probably, with some non-negative weight function near the simple partial fraction Website: https://zoom.us/j/98008001815?pwd=OG1rTVRFRzFpY3RhZmE4MXFwckxMUT09 * ID: 980 0800 1815; Password: 055016 |