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Some inequalities for polynomials and rational functions associated with lemniscates

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Inequalities for the area of a lemniscate and inequalities for polynomials in a lemniscate containing no critical points distinct from zeros are presented. Also, multipoint distortion estimates on boundaries of lemniscates are proved. Some open problems are discussed. One of them is connected with the well-known Smale’s mean value conjecture. Bibliography: 16 titles.

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References

  1. P. Borwein and T. Erdelyi, Polynomials and Polynomial Inequalities (Grad. Texts Math., 161), Springer-Verlag, New York (1995).

  2. G. Pólya, “Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehrfach zusammenhéingende Gebiete,” S. B. Preuss. Akad. Wiss. Berlin, Math. -phys.- tech. Kl., 280–282 (1928).

  3. V. N Dubinin, “Inequalities for critical values of polynomials,” Mat. Sb., 197, No. 8, 63–72 (2006).

    Article  MathSciNet  Google Scholar 

  4. S. Smale, “The fundamental theorem of algebra and complexity theory,” Bull. Amer. Math. Soc., 4, 1–36 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  5. G. V. Kuz’mina, “Methods of geometric function theory, I, II,” Algebra Analiz, 9 No. 3, 41–103 (1997); 9, No. 5, 1–50 (1997).

  6. G. M. Goluzin, Geometric Theory of Functions of a Complex Variable [in Russian]. Moscow (1966).

  7. R. B. Burckel, D. E. Marshall, D. Minda, P. Poggi-Corradini, and T. J. Ransford, “Area, capacity and diameter versions of Schwarz’s lemma,” Conform. Geom. Dyn., 12, 133–152 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  8. V. N. Dubinin, Condenser Capacities and Symmetrization in the Geometric Theory of Functions of a Complex Variable [in Russian], Vladivostok (2009).

  9. I. P. Mityuk, “Symmetrization principle for multiply connected domains,” Dokl. AN SSSR, 157, 268–270 (1964).

    Google Scholar 

  10. G. P. Bakhtina, “On an extremal problem of the conformal map of a unit disk onto nonoverlapping domains,” Urk. Mat. Zh., 26, 635–637 (1974).

    Google Scholar 

  11. E. Fujikawa and T. Sugawa, “Geometric function theory and Smale’s mean value conjecture,” Proc. Japan Acad. Ser. A, Math. Sci., 82, No. 7, 97–100 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  12. B. Sendov and P. Marinov, “Verification of Smale’s mean value conjecture for n ≤ 10,” C. R. Acad. Bulgare Sci., 60, No. 11, 1151–1156 (2007).

    MathSciNet  MATH  Google Scholar 

  13. E. Crane, “A bound for Smale’s mean value conjecture for complex polynomials,” Bull. London Math. Soc., 39, 781–791 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  14. Y. Wang, “Some results on Smale’s mean value conjecture,” in: Third International Congress of Chinese Mathematicians, Part 2 (AMS/IP Stud. Adv. Math., 42), Amer. Math. Soc., Providence, Rhode Island (2008), pp. 595–602.

  15. A. Hinkkanen and I. Kayumov, “Smale’s problem for critical points on certain two rays,” J. Austr. Math. Soc., 88. No. 2. 183–191 (2010).

    Google Scholar 

  16. Yu. E. Alenitzyn. “On univalent functions without common values in a multiply connected domain,” Tr. Mat. Inst. AN SSSR, 94. 4–18 (1968).

    Google Scholar 

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Authors and Affiliations

  1. Far Eastern Federal University, Vladivostok, Russia

    V. N. Dubinin

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  1. V. N. Dubinin
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Correspondence to V. N. Dubinin.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 404, 2012, pp. 83–99.

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Dubinin, V.N. Some inequalities for polynomials and rational functions associated with lemniscates. J Math Sci 193, 45–54 (2013). https://doi.org/10.1007/s10958-013-1432-4

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  • DOI: https://doi.org/10.1007/s10958-013-1432-4

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