Growth of a polynomial not vanishing in a disk

Abdullah Mir

Abstract


This paper deals with the problem of finding some upper bound estimates for the maximum modulus of the derivative and higher order derivatives of a complex polynomial on a disk under the assumption that the polynomial has no zeros in another disk. The estimates obtained strengthen the well-known inequality of Ankeny and Rivlin about polynomials.

Keywords


Polynomial; maximum modulus principle; zeros

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References


Ankeny, N. C., Rivlin, T. J., On a theorem of S. Bernstein, Pacific J. Math. 5 (1955), 849–852.

Aziz, A., Aliya, Q., Growth of polynomials not vanishing in a disk of prescribed radius, Int. J. Pure Appl. Math. 41 (2007), 713–734.

Govil, N. K., Qazi, M. A., Rahman, Q. I., Inequalities describing the growth of polynomials not vanishing in a disk of prescribed radius, Math. Ineq. Appl. 6 (2003), 453–467.

Jain, V. K., A generalization of Ankeny and Rivlin’s result on the maximum modulus of polynomials not vanishing in the interior of the unit circle, Turk. J. Math. 31 (2007), 89–94.

Milovanovic, G. V., Mitrinovic, D. S., Rassias, Th. M., Topics in polynomials, Extremal problems, Inequalities, Zeros, World scientific, Singapore, 1944.




DOI: http://dx.doi.org/10.17951/a.2019.73.1.41-48
Date of publication: 2019-12-19 10:33:47
Date of submission: 2019-12-17 10:02:17


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