Geodesic
Polynomials with Critical Values on Intervals
Matematičeskie zametki, Tome 78 (2005) no. 6, pp. 827-832

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For polynomials $P(z)$ with real coefficients having a fixed leading coefficient and satisfying the conditions $P(z)\in[-1,1]$ for $z\in[-1,1]$ and $P(z)\in[-1,1]$ if $P'(z)=0$, we obtain new covering theorems, a Bernshtein-type inequality, and inequalities for the coefficients. The proofs are based on the use of univalent conformal mappings. 
@article{MZM_2005_78_6_a2,
     author = {V. N. Dubinin},
     title = {Polynomials with {Critical} {Values} on {Intervals}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {827--832},
     publisher = {mathdoc},
     volume = {78},
     number = {6},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2005_78_6_a2/}
}
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V. N. Dubinin. Polynomials with Critical Values on Intervals. Matematičeskie zametki, Tome 78 (2005) no. 6, pp. 827-832. http://geodesic.mathdoc.fr/item/MZM_2005_78_6_a2/