Geodesic
Some inequalities for polynomials and rational functions associated with a~lemniscate
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 27, Tome 404 (2012), pp. 83-99

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Inequalities for the area of a lemniscate, and inequalities for polynomials in a lemniscate containing no critical points other than zeros are given. Also, multipoint distortion estimates at the boundary of a lemniscate are proved. Some open problems are discussed. One of them associates with the well-known Smale's mean value conjecture. 
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     author = {V. N. Dubinin},
     title = {Some inequalities for polynomials and rational functions associated with a~lemniscate},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {83--99},
     publisher = {mathdoc},
     volume = {404},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_404_a4/}
}
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V. N. Dubinin. Some inequalities for polynomials and rational functions associated with a~lemniscate. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 27, Tome 404 (2012), pp. 83-99. http://geodesic.mathdoc.fr/item/ZNSL_2012_404_a4/