International Journal of Mathematics and Mathematical Sciences
Volume 28 (2001), Issue 4, Pages 231-235

Integral mean estimates for polynomials whose zeros are within a circle

K. K. Dewan, Abdullah Mir, and R. S. Yadav

Department of Mathematics, Faculty of Natural Sciences, Jamia Millia Islamia, New Delhi 110025, India

Received 3 November 2000

Copyright © 2001 K. K. Dewan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let p(z) be a polynomial of degree n having all its zeros in |z|k;k1, then for each r>0, p>1, q>1 with p1+q1=1, Aziz and Ahemad (1996) recently proved that n{02π|p(eiθ)|rdθ}1/r{02π|1+keiθ|prdθ}1/pr{02π|p(eiθ)|qrdθ}1/qr. In this paper, we extend the above inequality to the class of polynomials p(z)=anzn+v=μnanvznv;1μn having all its zeros in |z|k;k1 and obtain a generalization as well as a refinement of the above result.