Some inequalities of algebraic polynomials
Author:
A. K. Varma
Journal:
Proc. Amer. Math. Soc. 123 (1995), 20412048
MSC:
Primary 26D05; Secondary 41A10
MathSciNet review:
1231305
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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 Markovtype 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), 5882.
 [2]
P.
Erdös, On extremal properties of the derivatives of
polynomials, Ann. of Math. (2) 41 (1940),
310–313. MR 0001945
(1,323g)
 [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, http://dx.doi.org/10.1215/S0012709437003612
 [4]
G.
G. Lorentz, Derivatives of polynomials with positive
coefficients, J. Approximation Theory 1 (1968),
1–4. MR
0231957 (38 #283)
 [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 YorkLondon, 1980,
pp. 881–887. MR 602815
(82b:26017)
 [6]
G. Szegö, On some properties of approximation, Magyar Tud. Akad. Mat. Kutató Int. Közl. 2 (1964), 39.
 [7]
P.
Turan, Über die Ableitung von Polynomen, Compositio Math.
7 (1939), 89–95 (German). MR 0000228
(1,37b)
 [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
(84d:41024), http://dx.doi.org/10.1090/S00029939198306952485
 [9]
A.
K. Varma, Derivatives of polynomials with
positive coefficients, Proc. Amer. Math.
Soc. 83 (1981), no. 1, 107–112. MR 619993
(82j:26014), http://dx.doi.org/10.1090/S00029939198106199930
 [1]
 J. Eröd, Bizonyos polinomok maximumáról, Mat. Fiz. Lapok 46 (1939), 5882.
 [2]
 P. Erdös, Extremal properties of derivatives of polynomials, Ann. of Math. 2 (1940), 310313. MR 0001945 (1:323g)
 [3]
 E. Hille, G. Szegö, and J. D. Tamarkin, On some generalizations of a theorem of A. A. Markoff, Duke Math. J. 3 (1937), 729739. MR 1546027
 [4]
 G. G. Lorentz, Derivatives of polynomials with positive coefficients, J. Approx. Theory 1 (1968), 14. MR 0231957 (38:283)
 [5]
 J. Szabados and A. K. Varma, Inequalities for derivatives of polynomials having real zeros, Approximation Theory III, The University of Texas, Austin (E. W. Cheney, eds.), Academic Press, New York, 1980, pp. 881888. MR 602815 (82b:26017)
 [6]
 G. Szegö, On some properties of approximation, Magyar Tud. Akad. Mat. Kutató Int. Közl. 2 (1964), 39.
 [7]
 P. Turán, Über die Ableitung von Polynomen, Compositio Math. (1939), 8995. MR 0000228 (1:37b)
 [8]
 A. K. Varma, Some inequalities of algebraic polynomials having all zeros inside , Proc. Amer. Math. Soc. 88 (1983), 227233. MR 695248 (84d:41024)
 [9]
 , Derivative of polynomials with positive coefficients, Proc. Amer. Math. Soc. 83 (1981), 107112. MR 619993 (82j:26014)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199512313050
PII:
S 00029939(1995)12313050
Article copyright:
© Copyright 1995
American Mathematical Society
