Geodesic
Application of conformal mappings to the inequalities for polynomials
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 18, Tome 286 (2002), pp. 85-102

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Applications of the geometric theory of functions to inequalities for algebraic polynomials are considered. The main attention is paid to constructing a univalent conformal mapping for a given polynomial and to applying the Lebedev and Nehari theorems to this mapping. A new sharp inequality of Bernshtein type for polynomials with restrictions on the growth on a segment or on a circle, inequalities with restrictions on the zeros of the polynomial, and other inequalities are obtained. In particular, classical inequalities by Markov, Bernshtein, and Schur are strengthened. 
@article{ZNSL_2002_286_a6,
     author = {V. N. Dubinin and A. V. Olesov},
     title = {Application of conformal mappings to the inequalities for polynomials},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {85--102},
     publisher = {mathdoc},
     volume = {286},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2002_286_a6/}
}
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V. N. Dubinin; A. V. Olesov. Application of conformal mappings to the inequalities for polynomials. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 18, Tome 286 (2002), pp. 85-102. http://geodesic.mathdoc.fr/item/ZNSL_2002_286_a6/