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.

## References

P. Borwein and T. Erdelyi,

*Polynomials and Polynomial Inequalities*(Grad. Texts Math.,**161**), Springer-Verlag, New York (1995).G. Pólya, “Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehrfach zusammenhéingende Gebiete,”

*S. B. Preuss. Akad. Wiss. Berlin, Math. -phys.- tech. Kl*., 280–282 (1928).V. N Dubinin, “Inequalities for critical values of polynomials,”

*Mat. Sb.*,**197**, No. 8, 63–72 (2006).S. Smale, “The fundamental theorem of algebra and complexity theory,”

*Bull. Amer. Math. Soc*.,**4**, 1–36 (1981).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).G. M. Goluzin,

*Geometric Theory of Functions of a Complex Variable*[in Russian]. Moscow (1966).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).V. N. Dubinin,

*Condenser Capacities and Symmetrization in the Geometric Theory of Functions of a Complex Variable*[in Russian], Vladivostok (2009).I. P. Mityuk, “Symmetrization principle for multiply connected domains,”

*Dokl. AN SSSR*,**157**, 268–270 (1964).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).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).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).E. Crane, “A bound for Smale’s mean value conjecture for complex polynomials,”

*Bull. London Math. Soc.*,**39**, 781–791 (2007).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.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).Yu. E. Alenitzyn. “On univalent functions without common values in a multiply connected domain,”

*Tr. Mat. Inst. AN SSSR*,**94**. 4–18 (1968).

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

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

## Rights and permissions

## About this article

### Cite this article

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

Received:

Published:

Issue Date:

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

### Keywords

- Rational Function
- Open Problem
- Distortion Estimate