A comparison inequality for rational functions

Author:
Xin Li

Journal:
Proc. Amer. Math. Soc. **139** (2011), 1659-1665

MSC (2010):
Primary 26D10, 26Cxx; Secondary 30A10, 30C15

Published electronically:
September 16, 2010

MathSciNet review:
2763755

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Abstract | References | Similar Articles | Additional Information

Abstract: We establish a new inequality for rational functions and show that it implies many inequalities for polynomials and their polar derivatives.

**1.**Abdul Aziz,*Inequalities for the polar derivative of a polynomial*, J. Approx. Theory**55**(1988), no. 2, 183–193. MR**965215**, 10.1016/0021-9045(88)90085-8**2.**Abdul Aziz and W. M. Shah,*Inequalities for the polar derivative of a polynomial*, Indian J. Pure Appl. Math.**29**(1998), no. 2, 163–173. MR**1623254****3.**Peter Borwein and Tamás Erdélyi,*Polynomials and polynomial inequalities*, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR**1367960****4.**Peter Borwein and Tamás Erdélyi,*Sharp extensions of Bernstein’s inequality to rational spaces*, Mathematika**43**(1996), no. 2, 413–423 (1997). MR**1433285**, 10.1112/S0025579300011876**5.**F. F. Bonsall and Morris Marden,*Critical points of rational functions with self-inversive polynomial factors*, Proc. Amer. Math. Soc.**5**(1954), 111–114. MR**0060016**, 10.1090/S0002-9939-1954-0060016-6**6.**N. K. Govil,*On the derivative of a polynomial*, Proc. Amer. Math. Soc.**41**(1973), 543–546. MR**0325932**, 10.1090/S0002-9939-1973-0325932-8**7.**N. K. Govil, Griffith Nyuydinkong, and Berhanu Tameru,*Some 𝐿^{𝑝} inequalities for the polar derivative of a polynomial*, J. Math. Anal. Appl.**254**(2001), no. 2, 618–626. MR**1805528**, 10.1006/jmaa.2000.7267**8.**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**, 10.1090/S0002-9904-1944-08177-9**9.**Xin Li, R. N. Mohapatra, and R. S. Rodriguez,*Bernstein-type inequalities for rational functions with prescribed poles*, J. London Math. Soc. (2)**51**(1995), no. 3, 523–531. MR**1332889**, 10.1112/jlms/51.3.523**10.**M. A. Malik and M. C. Vong,*Inequalities concerning the derivative of polynomials*, Rend. Circ. Mat. Palermo (2)**34**(1985), no. 3, 422–426 (1986). MR**848119**, 10.1007/BF02844535**11.**Morris Marden,*Geometry of polynomials*, Second edition. Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR**0225972****12.**R.N. Mohapatra and W.M. Shah,*Inequalities for the polar derivative of a polynomial*, preprint, 2008.**13.**Q. I. Rahman and G. Schmeisser,*Analytic theory of polynomials*, London Mathematical Society Monographs. New Series, vol. 26, The Clarendon Press, Oxford University Press, Oxford, 2002. MR**1954841****14.**V.I. Smirnov and N.A. Lebedev,*Function of a Complex Variable*, English edition, Iliffe Books, Landon, 1968.

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Additional Information

**Xin Li**

Affiliation:
Department of Mathematics, University of Central Florida, Orlando, Florida 32816

Email:
xli@math.ucf.edu

DOI:
https://doi.org/10.1090/S0002-9939-2010-10624-X

Keywords:
Bernstein inequality,
rational functions,
polar derivative

Received by editor(s):
February 4, 2010

Received by editor(s) in revised form:
May 17, 2010

Published electronically:
September 16, 2010

Communicated by:
Walter Van Assche

Article copyright:
© Copyright 2010
American Mathematical Society

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