Abstract
Let
be a polynomial of degree n and let M(f, r) = \( \mathop {\max }\limits_{\left| z \right| = r} \left| {f\left( z \right)} \right| \) for an arbitrary entire function f(z). If P(z) has no zeros in |z| < 1 with M(P,1) = 1, then for |α| ≤ 1, it is proved by Jain[Glasnik Matematički, 32(52) (1997), 45–51] that
.
In this paper, we shall first obtain a result concerning minimum modulus of polynomials and next improve the above inequality for polynomials with restricted zeros. Our result improves the well known inequality due to Ankeny and Rivlin[1] and besides generalizes some well known polynomial inequalities proved by Aziz and Dawood[3].
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References
Ankeny, N. C. and Rivlin, T. J., On a Theorem of S. Bernstein, Pacific J. Math., 5(1955), 849–852.
Aziz, A., Growth of Polynomials whose Zeros are Within or Outside a Circle, Bull. Austral. Math. Soc., 35(1987), 247–256.
Aziz, A. and Dawood, Q. M., Inequalities for a Polynomial and its Derivative, J. Approx. Theory, 54(1988), 306–313.
Jain, V. K., On Maximum Modulus of Polynomial, Indian J. Pure and Applied Math., 23:11(1992), 815–819.
Jain, V. K., Generalization of Certain well Known Inequalities for Polynomials, Glasnik Matematički, 32:52(1997), 45–51.
Pólya, G. and Szegö, G., Aufgaben und Lehrsatze aus der Analysis, Springer-Verlag, Berlin, (1925).
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Supported by Council of Scientific and Industrial Research, New Delhi, under grant F.No. 9/466(95)/2007-EMR-I.
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Dewan, K.K., Hans, S. Some polynomial inequalities in the complex domain. Anal. Theory Appl. 26, 1–6 (2010). https://doi.org/10.1007/s10496-010-0001-7
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DOI: https://doi.org/10.1007/s10496-010-0001-7