Abstract
For a polynomial P(z) of degree n which has no zeros in z < 1, Dewan et al. [4] established the inequality
for any β ≤ 1 and z = 1. In this paper we improve the above inequality for the sth derivative of a polynomial which has no zeros in z < k, k ≤ 1. Our results generalize certain well-known polynomial inequalities.
абстрактный
Для многочлена P(z) степени п, не имеющего корней в круге z < 1, Деван и др. [4] установили, что
для любых ß < 1 и z = 1. В ѳтои статье мы уточняем неравенство выше для s-H производнои многочлена, не имеющего нулеи в z < к, к < 1. Наши результаты обобщают некоторые известные неравенства для многочленов.
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References
A. Aziz and Q. M. Dawood, Inequalities for a polynomial and its derivative, J. Approx. Theory, 54(1988), 306–313.
A. Aziz and Q. G. Mohammad, Growth of polynomial with zeros outside a circle, Proc. Amer. Math. Soc., 81(1981), 549–553.
S. BERNSTEIN, Leçons sur les propriétés extrémales et la meilleure approximation des fonctions analytiques dune variable réelle, Gauthier Villars (Paris, 1926).
K. K. Dewan and S. Hans, Generalization of certain well-known polynomial inequalities, J. Math. Anal. Appl., 363(2010), 38–41.
R. B. Gardner, N. K. GoviL and S. R. MüSüKüLA, Rate of growth of polynomials not vanishing inside a circle, J. Inequal. Pure Appl. Math., 6(2005), 1–9.
N. K. Govil, On a theorem of S. Bernstein, J. Math. Phys. Sci., 14(1980), 183–187.
N. K. Govil, Some inequalities for derivatives of polynomials, J. Approx. Theory, 66(1991), 29–35.
S. Hans and R. Lal, Generalization of some polynomial inequalities not vanishing in a disk, Analysis Math., 40(2014), 105–115.
V. K. Jain, Generalization of certain well-known inequalities for polynomials, Glasnik Mat., 32(1997), 45–51.
P. D. Lax, Proof of a conjecture of P. Erdos on the derivative of a polynomial, Bull. Amer. Math. Soc., 50(1944), 509–513.
M. A. Malik, On the derivative of a polynomial, J. London Math. Soc., 1(1969), 57–60.
Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press (New York, 2002).
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Zireh, A. Generalization of certain well-known inequalities for the derivative of polynomials. Anal Math 41, 117–132 (2015). https://doi.org/10.1007/s10476-015-0109-2
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DOI: https://doi.org/10.1007/s10476-015-0109-2