TY - JOUR
T1 - Some Sharpening and Generalizations of a Result of T. J. Rivlin
JO - Analysis in Theory and Applications
VL - 3
SP - 219
EP - 228
PY - 2017
DA - 2017/08
SN - 33
DO - http://doi.org/10.4208/ata.2017.v33.n3.3
UR - https://global-sci.org/intro/article_detail/ata/10513.html
KW - Inequalities, polynomials, zeros.
AB - <p style="text-align: justify;">Let $p(z)=a_0+a_1z+a_2z^2+a_3z^3+\cdots+a_nz^n$ be a polynomial of degree $n$. Rivlin [12] proved that if $p(z)\neq 0$ in the unit disk, then for $0&lt; r\leq 1,$ $${\max_{|z|=r}|p(z)|}\geq \Big(\dfrac{r+1}{2}\Big)^n{\max_{|z|=1}|p(z)|}.$$ In this paper, we prove a sharpening and generalization of this result and show by means of examples that for some polynomials our result can significantly improve the bound obtained by the Rivlin&#39;s Theorem.</p>