Integral mean estimates for polynomials whose zeros are within a circle

K. K. Dewan, Abdullah Mir and R. S. Yadav

International Journal of Mathematics and Mathematical Sciences, 2001, vol. 28, 1-5

Abstract:

Let p ( z ) be a polynomial of degree n having all its zeros in | z | ≤ k ; k ≤ 1 , then for each r > 0 , p > 1 , q > 1 with p − 1 + q − 1 = 1 , Aziz and Ahemad (1996) recently proved that n { ∫ 0 2 π | p ( e i θ ) | r d θ } 1 / r ≤ { ∫ 0 2 π | 1 + k e i θ | p r d θ } 1 / p r { ∫ 0 2 π | p ′ ( e i θ ) | q r d θ } 1 / q r . In this paper, we extend the above inequality to the class of polynomials p ( z ) = a n z n + ∑ v = μ n a n − v z n − v ; 1 ≤ μ ≤ n having all its zeros in | z | ≤ k ; k ≤ 1 and obtain a generalization as well as a refinement of the above result.

Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:916325

DOI: 10.1155/S016117120100607X

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