Some properties of self-inversive polynomials

Authors:
P. J. O’Hara and R. S. Rodriguez

Journal:
Proc. Amer. Math. Soc. **44** (1974), 331-335

MSC:
Primary 30A06

DOI:
https://doi.org/10.1090/S0002-9939-1974-0349967-5

MathSciNet review:
0349967

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Abstract | References | Similar Articles | Additional Information

Abstract: A complex polynomial is called self-inversive [**5**, p. 201] if its set of zeros (listing multiple zeros as many times as their multiplicity indicates) is symmetric with respect to the unit circle. We prove that if is self-inversive and of degree then where and denote the maximum modulus of and , respectively, on the unit circle. This extends a theorem of P. Lax [**4**]. We also show that if has all its zeros on then . Finally, as a consequence of this inequality, we show that when has all its zeros on then and for . This answers in part a question presented in [**3**, p. 24].

**[1]**Germán Ancochea,*Zeros of self-inversive polynomials*, Proc. Amer. Math. Soc.**4**(1953), 900–902. MR**0058748**, https://doi.org/10.1090/S0002-9939-1953-0058748-8**[2]**N. G. de Bruijn,*Inequalities concerning polynomials in the complex domain*, Nederl. Akad. Wetensch., Proc.**50**(1947), 1265–1272 = Indagationes Math. 9, 591–598 (1947). MR**0023380****[3]**W. K. Hayman,*Research problems in function theory*, The Athlone Press University of London, London, 1967. MR**0217268****[4]**Peter D. Lax,*Proof of a conjecture of P. Erdös on the derivative of a polynomial*, Bull. Amer. Math. Soc.**50**(1944), 509–513. MR**0010731**, https://doi.org/10.1090/S0002-9904-1944-08177-9**[5]**Morris Marden,*Geometry of polynomials*, Second edition. Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR**0225972**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1974-0349967-5

Keywords:
Self-inversive polynomials,
coefficients,
maximum modulus,
inequalities

Article copyright:
© Copyright 1974
American Mathematical Society