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Journal of Zhejiang University SCIENCE A 2005 Vol.6 No.7 P.764~768

http://doi.org/10.1631/jzus.2005.A0764


On Turán type inequality with doubling weights and A* weights


Author(s):  YU Dan-sheng, WEI Bao-rong

Affiliation(s):  Department of Mathematics, Zhejiang University, Hangzhou 310028, China; more

Corresponding email(s):   yudansheng@sina.com

Key Words:  Turá, n type inequality, Doubling weights, A* weights


YU Dan-sheng, WEI Bao-rong. On Turán type inequality with doubling weights and A* weights[J]. Journal of Zhejiang University Science A, 2005, 6(7): 764~768.

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author="YU Dan-sheng, WEI Bao-rong",
journal="Journal of Zhejiang University Science A",
volume="6",
number="7",
pages="764~768",
year="2005",
publisher="Zhejiang University Press & Springer",
doi="10.1631/jzus.2005.A0764"
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%A WEI Bao-rong
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%I Zhejiang University Press & Springer
%DOI 10.1631/jzus.2005.A0764

TY - JOUR
T1 - On Turán type inequality with doubling weights and A* weights
A1 - YU Dan-sheng
A1 - WEI Bao-rong
J0 - Journal of Zhejiang University Science A
VL - 6
IS - 7
SP - 764
EP - 768
%@ 1673-565X
Y1 - 2005
PB - Zhejiang University Press & Springer
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DOI - 10.1631/jzus.2005.A0764


Abstract: 
Let Hn be the set of real algebraic polynomials of degree n, whose zeros all lie in the interval [-1,1]. The well known turá;n type inequalities tell us that for f(x)∈Hn, it holds ‖f′‖≥Cnf‖. This note deals with the weighted turá;n type inequalities with the weights having inner singularities under Lp norm for 0<p≤∞. Our results essentially extend the result of Wang and Zhou (2002), and the method used in this paper is simpler and more direct than that of Wang and Zhou (2002). The results and methods have their own values in approximation theory and computation.

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article

Reference

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[3] Erdélyi, T., 1999. Notes on inequalities with doubling weights. J. Approx. Theory, 100:60-72.

[4] Eröd, J., 1939. Bizonyos polinomok maximumairćl. Math. Fiz. Lapok, 46:58-82.

[5] Mastroianni, G., Totik, V., 2000. Weighted polynomial inequalities with doubling weights and A weights. Constr. Approx., 16:37-71.

[6] Mastroianni, G., Totik, V., 2001. Best approximation and moduli of smoothness for doubling weights. J. Approx. Theory, 110:180-199.

[7] Milovanović, G.V., Rassias, T.M., 1998. New Development on Turán Extremal Problems for Polynomials. Approximation Theory: In Memory of A. K. Varma. Marcel Dekker Inc., New York, p.433-447.

[8] Turán, P., 1939. Uber dir ableitung von polynomen. Compositio Math., 7:89-95.

[9] Varma, A.K., 1976. An analogue of some inequality of P.Turán concerning algebraic polynomials satisfying certain conditions. Proc. Amer. Math. Soc., 55:305-309.

[10] Varma, A.K., 1978. An analogue of some inequality of P.Turán concerning algebraic polynomials having all zeros inside [(1,1]. Proc. Amer. Math. Soc., 69:25-33.

[11] Varma, A.K., 1983. Some inequalities of algebraic polynomials having all zeros inside [(1,1]. Proc. Amer. Math. Soc., 88:227-233.

[12] Wang, J.L., Zhou, S.P., 2002. The weighted Turán typeinequality for generalized Jacobi weights. Bull. Austral. Math. Soc., 66:259-265.

[13] Xiao, W., Zhou, S.P., 1999. On weighted Turán type inequality. Glasnik Math., 54:197-202.

[14] Zhou, S.P., 1992. Some remarks on Turán type inequality. J. Approx. Theory, 68:45-48.

[15] Zhou, S.P., 1993. Some remarks on Turán type inequality II. J. Math. Anal. Appl., 180:138-143.

[16] Zhou, S.P., 1995. Some remarks on Turán type inequality III: the completion. Anal. Math., 21:313-318.

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