Some inequalities of algebraic polynomials

Author:
A. K. Varma

Journal:
Proc. Amer. Math. Soc. **123** (1995), 2041-2048

MSC:
Primary 26D05; Secondary 41A10

DOI:
https://doi.org/10.1090/S0002-9939-1995-1231305-0

MathSciNet review:
1231305

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Erdös and Lorentz showed that by considering the special kind of the polynomials better bounds for the derivative are possible. Let us denote by the set of all polynomials whose degree is *n* and whose zeros are real and lie inside . Let and ; then the object of Theorem 1 is to obtain the best lower bound of the expression for and characterize the polynomial which achieves this lower bound. Next, we say that if is a polynomial whose degree is *n* and whose roots are all real and do not lie inside . In Theorem 2, we shall prove Markov-type inequality for such a class of polynomials belonging to in the weighted norm (*p* integer). Here . In Theorem 3 we shall consider another analogous problem as in Theorem 2.

**[1]**J. Eröd,*Bizonyos polinomok maximumáról*, Mat. Fiz. Lapok**46**(1939), 58-82.**[2]**P. Erdös,*On extremal properties of the derivatives of polynomials*, Ann. of Math. (2)**41**(1940), 310–313. MR**0001945**, https://doi.org/10.2307/1969005**[3]**Einar Hille, G. Szegö, and J. D. Tamarkin,*On some generalizations of a theorem of A. Markoff*, Duke Math. J.**3**(1937), no. 4, 729–739. MR**1546027**, https://doi.org/10.1215/S0012-7094-37-00361-2**[4]**G. G. Lorentz,*Derivatives of polynomials with positive coefficients*, J. Approximation Theory**1**(1968), 1–4. MR**0231957****[5]**Josef Szabados and A. K. Varma,*Inequalities for derivatives of polynomials having real zeros*, Approximation theory, III (Proc. Conf., Univ. Texas, Austin, Tex., 1980), Academic Press, New York-London, 1980, pp. 881–887. MR**602815****[6]**G. Szegö,*On some properties of approximation*, Magyar Tud. Akad. Mat. Kutató Int. Közl.**2**(1964), 3-9.**[7]**P. Turan,*Über die Ableitung von Polynomen*, Compositio Math.**7**(1939), 89–95 (German). MR**0000228****[8]**A. K. Varma,*Some inequalities of algebraic polynomials having all zeros inside [-1,1]*, Proc. Amer. Math. Soc.**88**(1983), no. 2, 227–233. MR**695248**, https://doi.org/10.1090/S0002-9939-1983-0695248-5**[9]**A. K. Varma,*Derivatives of polynomials with positive coefficients*, Proc. Amer. Math. Soc.**83**(1981), no. 1, 107–112. MR**619993**, https://doi.org/10.1090/S0002-9939-1981-0619993-0

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
26D05,
41A10

Retrieve articles in all journals with MSC: 26D05, 41A10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1995-1231305-0

Article copyright:
© Copyright 1995
American Mathematical Society