Geodesic
Extremal problems for algebraic polynomials
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 60 (2005) no. 6, pp. 1183-1194 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $$ [L]=\sup\bigl\{\min\{|L(p)(z_k)|:k=1,2,\dots,n\}:z_1z_2\dotsb z_n=1\bigr\} $$ 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 $$ L(p)(z)=L\biggl(\sum_{k=0}^na_kz^k\biggr)=\sum_{k=0}^n\frac1{k+1}a_kz^k=\frac1z\int_0^zp(u)\,du, \enskip a_0=1, \ \ a_n=(-1)^n. $$ 
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     author = {B. Kh. Sendov},
     title = {Extremal problems for algebraic polynomials},
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     url = {http://geodesic.mathdoc.fr/item/RM_2005_60_6_a10/}
}
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B. Kh. Sendov. Extremal problems for algebraic polynomials. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 60 (2005) no. 6, pp. 1183-1194. http://geodesic.mathdoc.fr/item/RM_2005_60_6_a10/

[1] A. F. Beardon, D. Minda, T. W. Ng, “Smale's mean value conjecture and the hyperbolic metric”, Math. Ann., 322:4 (2002), 623–632 | DOI | MR | Zbl

[2] Q. I. Rahman, G. Schmeisser, Analytic Theory of Polynomials, Oxford Univ. Press, Clarendon Press, Oxford, 2002 | MR | Zbl

[3] Bl. Sendov, N. Nikolov, “Min problems for complex polynomials”, East J. Approx., 9:4 (2003), 427–442 | MR | Zbl

[4] S. Smale, “The fundamental theorem of algebra and complexity theory”, Bull. Amer. Math. Soc., 4:1 (1981), 1–36 | DOI | MR | Zbl

[5] D. Tischler, “Critical points and values of complex ploynomials”, J. Complexity, 5:4 (1989), 438–456 | DOI | MR | Zbl

[6] D. Tischler, “Perturbations of critical fixed points of analytic mapes”, Astérisque, 222 (1994), 407–422 | MR | Zbl

[7] J. Tyson, “Conterexamples to Tischler's strong form of Smale's mean value conjecture”, Bull. London Math. Soc., 37:1 (2005), 95–106 | DOI | MR | Zbl