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 $$M(P',\rho)\leq\frac{n}{\rho+k}\{(\frac{\rho+k}{r+k})^n[1-\frac{(k-\rho)(n|a_0|-k|a_1|)n}{(\rho^2+k^2)n|a_0|+2k^2\rho |a_1|}(\frac{\rho-r}{k+r})(\frac{k+1}{k+\rho})^{n-1}]M(P,r)$$ $$-[\frac{(n|a_0|\rho+k^2|a_1|)(r+k)}{(\rho^2+k^2)n|a_0|+2k^2\rho|a_1|}\times[((\frac{\rho+k}{r+k})^n-1)-n(\rho-r)]]m(P,k)\}$$ In this paper, we obtain a refinement of the above inequality. Moreover, we obtain a generalization of above inequality for $M(P', R)$, where $R\geq k$.
Zireh,A . (2011). Maximum modulus of derivatives of a polynomial. International Journal of Nonlinear Analysis and Applications, 2(2), 109-113. doi: 10.22075/ijnaa.2011.106
MLA
Zireh,A . "Maximum modulus of derivatives of a polynomial", International Journal of Nonlinear Analysis and Applications, 2, 2, 2011, 109-113. doi: 10.22075/ijnaa.2011.106
HARVARD
Zireh A. (2011). 'Maximum modulus of derivatives of a polynomial', International Journal of Nonlinear Analysis and Applications, 2(2), pp. 109-113. doi: 10.22075/ijnaa.2011.106
CHICAGO
A Zireh, "Maximum modulus of derivatives of a polynomial," International Journal of Nonlinear Analysis and Applications, 2 2 (2011): 109-113, doi: 10.22075/ijnaa.2011.106
VANCOUVER
Zireh A. Maximum modulus of derivatives of a polynomial. IJNAA. 2011;2(2):109-113. doi: 10.22075/ijnaa.2011.106
International Journal of Nonlinear Analysis and Applications