Geodesic
Extremal problems for algebraic polynomials
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 60 (2005) no. 6, pp. 1183-1194 
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/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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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