Integral Mean Estimates for Polynomials Whose Zeros are within a Circle ()
Abstract
Let be a polynomial of degree n having all its zeros in , then for each , ,
with , Aziz and Ahemad (1996) proved that
In this paper, we extend the above inequality to the class of polynomials , having all its zeros in , and obtain a generalization as well as refinement of the above result.
Share and Cite:
Paul, Y. , Shah, W. and Singh, G. (2011) Integral Mean Estimates for Polynomials Whose Zeros are within a Circle.
Applied Mathematics,
2, 141-144. doi:
10.4236/am.2011.21016.
Conflicts of Interest
The authors declare no conflicts of interest.
References
|
[1]
|
P. Turan, “über die Ableitung von Polynomen,” Compositio Mathematica, Vol. 7, 1939, pp. 89-95.
|
|
[2]
|
M. A. Malik, “On the Derivative of a Polynomial,” Journal of the London Mathematical Society, Vol. 2, No. 1, 1969, pp. 57-60. doi:10.1112/jlms/s2-1.1.57
|
|
[3]
|
M. A. Malik, “An Integral Mean Estimate for Polynomials,” Proceedings of the American Mathematical Society, Vol. 91, No. 2, 1984, pp. 281-284.
|
|
[4]
|
A. Aziz and W. M. Shah, “An Integral Mean Estimate for Polynomials,” Indian Journal of Pure and Applied Mathematics, Vol. 28, No. 10, 1997, pp. 1413-1419.
|
|
[5]
|
A. Aziz and N. Ahemad, “Integral Mean Estimates for Polynomials Whose Zeros Are within a Circle,” Glasnik Matematicki Series, III, Vol. 31, No. 2, 1996, pp. 229-237.
|
|
[6]
|
M. A. Qazi, “On the Maximum Modulus of Polynomials,” Proceedings of the American Mathematical Society, Vol. 115, 1990, pp. 337-343. doi:10.1090/S0002-9939-1992-1113648-1
|
|
[7]
|
N. K. Govil and G. N. McTume, “Some Generalizations Involving the Polar Derivative for an Inequality of Paul Turan,” Acta Mathematica Thunder, Vol. 104, No. 1-2, 2004, pp. 115-126. doi:10.1023/B:AMHU.0000034366. 66170.01
|
|
[8]
|
E. Hille, “Analytic Function Theory, Vol. II,” Introduction to Higher Mathematics, Ginn and Company, New York, 1962.
|