Document Type : Research Paper

Author

• A. Zireh

Department of Mathematics, Shahrood University of Technology, Shahrood, Iran.

Abstract

For an arbitrary entire function $f(z)$, let $M(f,R)=\underset{|z|=R}{max}|f(z)|$ and $m(f,r)=\underset{|z|=r}{min}|f(z)|$. If $P(z)$ is a polynomial of degree $n$ having no zeros in $|z|<k,k\ge 1$, then for $0\le r\le \rho \le k$, it is proved by Aziz et al. that
$$M(P^{\prime },\rho )\le \frac{n}{\rho +k}\{(\frac{\rho +k}{r+k}{)}^n[1-\frac{(k-\rho )(n|a_0|-k|a_1|)n}{({\rho }^2+k^2)n|a_0|+2k^2\rho |a_1|}(\frac{\rho -r}{k+r})(\frac{k+1}{k+\rho }{)}^{n-1}]M(P,r)$$
$$-[\frac{(n|a_0|\rho +k^2|a_1|)(r+k)}{({\rho }^2+k^2)n|a_0|+2k^2\rho |a_1|}\times [((\frac{\rho +k}{r+k}{)}^n-1)-n(\rho -r)]]m(P,k)\}$$
In this paper, we obtain a refinement of the above inequality. Moreover, we obtain
a generalization of above inequality for $M(P^{\prime },R)$, where $R\ge k$.

Keywords

• Polynomial
• inequality
• Maximum modulus
• Restricted Zeros