A quadratic approximation to the Sendov radius near the unit circle

Author:
Michael J. Miller

Journal:
Trans. Amer. Math. Soc. **357** (2005), 851-873

MSC (2000):
Primary 30C15

DOI:
https://doi.org/10.1090/S0002-9947-04-03766-3

Published electronically:
October 19, 2004

MathSciNet review:
2110424

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Define to be the set of complex polynomials of degree with all roots in the unit disk and at least one root at . For a polynomial , define to be the distance between and the closest root of the derivative . Finally, define . In this notation, a conjecture of Bl. Sendov claims that .

In this paper we investigate Sendov's conjecture near the unit circle, by computing constants and (depending only on ) such that for near . We also consider some consequences of this approximation, including a hint of where one might look for a counterexample to Sendov's conjecture.

**1.**B. D. Bojanov, Q. I. Rahman, and J. Szynal,*On a conjecture of Sendov about the critical points of a polynomial*, Math. Z.**190**(1985), no. 2, 281–285. MR**797543**, https://doi.org/10.1007/BF01160464**2.**B. D. Bojanov, Q. I. Rahman, and J. Szynal,*On a conjecture about the critical points of a polynomial*, Delay equations, approximation and application (Mannheim, 1984) Internat. Schriftenreihe Numer. Math., vol. 74, Birkhäuser, Basel, 1985, pp. 83–93. MR**899090****3.**Iulius Borcea,*The Sendov conjecture for polynomials with at most seven distinct zeros*, Analysis**16**(1996), no. 2, 137–159. MR**1397576**, https://doi.org/10.1524/anly.1996.16.2.137**4.**Johnny E. Brown and Guangping Xiang,*Proof of the Sendov conjecture for polynomials of degree at most eight*, J. Math. Anal. Appl.**232**(1999), no. 2, 272–292. MR**1683144**, https://doi.org/10.1006/jmaa.1999.6267**5.**Morris Marden,*Geometry of polynomials*, Second edition. Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR**0225972****6.**Michael J. Miller,*Maximal polynomials and the Ilieff-Sendov conjecture*, Trans. Amer. Math. Soc.**321**(1990), no. 1, 285–303. MR**965744**, https://doi.org/10.1090/S0002-9947-1990-0965744-X**7.**Michael J. Miller,*On Sendov’s conjecture for roots near the unit circle*, J. Math. Anal. Appl.**175**(1993), no. 2, 632–639. MR**1219199**, https://doi.org/10.1006/jmaa.1993.1194**8.**Michael J. Miller,*Some maximal polynomials must be nonreal*, J. Math. Anal. Appl.**214**(1997), no. 1, 283–291. MR**1645480**, https://doi.org/10.1006/jmaa.1997.5629**9.**Q. I. Rahman,*On the zeros of a polynomial and its derivative*, Pacific J. Math.**41**(1972), 525–528. MR**308374****10.**Zalman Rubinstein,*On a problem of Ilyeff*, Pacific J. Math.**26**(1968), 159–161. MR**237753****11.**J. V. Uspensky,*Theory of Equations*, McGraw-Hill, New York, 1948.**12.**V. Vâjâitu and A. Zaharescu,*Ilyeff’s conjecture on a corona*, Bull. London Math. Soc.**25**(1993), no. 1, 49–54. MR**1190363**, https://doi.org/10.1112/blms/25.1.49

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
30C15

Retrieve articles in all journals with MSC (2000): 30C15

Additional Information

**Michael J. Miller**

Affiliation:
Department of Mathematics, Le Moyne College, Syracuse, New York 13214

Email:
millermj@mail.lemoyne.edu

DOI:
https://doi.org/10.1090/S0002-9947-04-03766-3

Keywords:
Sendov,
Ilieff,
Ilyeff

Received by editor(s):
October 15, 2001

Published electronically:
October 19, 2004

Article copyright:
© Copyright 2004
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.