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Uspekhi Mat. Nauk, 2005, Volume 60, Issue 6(366), Pages 175–186 (Mi umn1682)  

Extremal problems for algebraic polynomials

B. Kh. Sendov

Bulgarian Academy of Sciences

Abstract: Let $L(p)$ be a linear operator on the set of monic algebraic polynomials $p(z)= (z_1-z)(z_2-z)\dotsb(z_n-z)$ with $z_1z_2\dotsb z_n=1$. Of interest here is the value
$$ [L]=\sup\{\min\{|L(p)(z_k)|:k=1,2,…,n\}:z_1z_2\dotsb z_n=1\} $$
for various linear operators. The motivation is that Smale's mean value conjecture may be formulated as $[L]=1-1/(n+1)$ for the linear operator
$$ L(p)(z)=L(\sum_{k=0}^na_kz^k)=\sum_{k=0}^n\frac1{k+1}a_kz^k=\frac1z\int_0^zp(u) du, \enskip a_0=1, a_n=(-1)^n. $$


DOI: https://doi.org/10.4213/rm1682

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English version:
Russian Mathematical Surveys, 2005, 60:6, 1183–1194

Bibliographic databases:

UDC: 517.5
MSC: 30C10
Received: 20.09.2005

Citation: B. Kh. Sendov, “Extremal problems for algebraic polynomials”, Uspekhi Mat. Nauk, 60:6(366) (2005), 175–186; Russian Math. Surveys, 60:6 (2005), 1183–1194

Citation in format AMSBIB
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