Maximum modulus of derivatives of a polynomial | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 12، دوره 2، شماره 2، 2011، صفحه 109-113 اصل مقاله (301.51 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2011.106 | ||
| نویسنده | ||
| A. Zireh* | ||
| Department of Mathematics, Shahrood University of Technology, Shahrood, Iran. | ||
| چکیده | ||
| 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 $R\geq k$. | ||
| کلیدواژهها | ||
| Polynomial؛ inequality؛ Maximum modulus؛ Restricted Zeros | ||
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