Integral mean estimates for polynomials whose zeros are within a circle

Abstract

Let p(z) be a polynomial of degree n having all its zeros in |z|≤k;  k≤1, then for each r>0, p>1, q>1 with p−1+q−1=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=μnan−vzn−v;1≤μ≤n having all its zeros in |z|≤k;  k≤1 and obtain a generalization as well as a refinement of the above result.