Inequalities concerning polar derivative of polynomials

Arty Ahuja, K. K. Dewan, Sunil Hans

Abstract


In this paper we obtain certain results for the polar derivative of a polynomial \(p(z) = c_nz^n +\sum_{j=\mu}^n c_{n-j}z^{n-j}\), \(1\leq\mu\leq n\), having all its zeros on \(|z| = k\), \(k\leq 1\), which generalizes the results due to Dewan and Mir, Dewan and Hans. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros.
[Editor's note: There are flaws in the paper, see M. A. Qazi, Remarks on some recent results about polynomials with restricted zeros, Ann. Univ. Mariae Curie-Skłodowska Sect. A 67 (2), (2013), 59-64 ]

Keywords


Polynomials; maximum modulus; inequalities in the complex domain; polar derivative

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References


Bernstein, S., Lecons sur les proprietes extremales et la meilleure approximation desfonctions analytiques d’une variable reele, Gauthier Villars, Paris, 1926 (French).

Chan, T. N., Malik, M. A., On Erdos-Lax theorem, Proc. Indian Acad. Sci. 92 (3) (1983), 191-193.

Dewan, K. K., Hans, S., On maximum modulus for the derivative of a polynomial, Ann. Univ. Mariae Curie-Skłodowska Sect. A 63 (2009), 55-62.

Dewan, K. K., Mir, A., Note on a theorem of S. Bernstein, Southeast Asian Bulletin of Math. 31 (2007), 691-695.

Govil, N. K., On the theorem of S. Bernstein, J. Math. Phys. Sci. 14 (1980), 183-187.

Govil, N. K., Rahman, Q. I., Functions of exponential type not vanishing in a half plane and related polynomials, Trans. Amer. Math. Soc. 137 (1969), 501-517.

Jain, V. K., On polynomials having zeros in closed exterior or interior of a circle, Indian J. Pure Appl. Math. 30 (1999), 153-159.

Malik, M. A., On the derivative of a polynomial, J. London Math. Soc. 1 (1969), 57-60.

Mir, A., On extremal properties and location of zeros of polynomials, Ph.D. Thesis submitted to Jamia Millia Islamia, New Delhi, 2002.

Qazi, M. A., On the maximum modulus of polynomials, Proc. Amer. Math. Soc. 115 (1992), 337-343.




DOI: http://dx.doi.org/10.17951/a.2011.65.1.1-9
Data publikacji: 2016-07-25 18:17:30
Data złożenia artykułu: 2016-07-25 14:20:08

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Copyright (c) 2011 Arty Ahuja, K. K. Dewan, Sunil Hans