Maximum modulus of derivatives of a polynomial


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


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