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Published in:
2016 | OriginalPaper | Chapter
Bernstein Type Inequalities Concerning Growth of Polynomials
Authors : N. K. Govil, Eze R. Nwaeze
Published in: Mathematical Analysis, Approximation Theory and Their Applications
Publisher: Springer International Publishing
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Abstract
Let \(p(z) = a_{0} + a_{1}z + a_{2}z^{2} + a_{3}z^{3} + \cdots + a_{n}z^{n}\) be a polynomial of degree n, where the coefficients a
j
, for 0 ≤ j ≤ n, may be complex, and p(z) ≠ 0 for | z | < 1. Then and where \(M(p,R):= \max _{\vert z\vert =R\geq 1}\vert p(z)\vert\), \(M(p,r):= \max _{\vert z\vert =r\leq 1}\vert p(z)\vert\), and \(\vert \vert p\vert \vert:= \max _{\vert z\vert =1}\vert p(z)\vert\). Inequality (1) is due to Ankeny and Rivlin (Pac. J. Math. 5, 849–852, 1955), whereas Inequality (2) is due to Rivlin (Am. Math. Mon. 67, 251–253, 1960). These inequalities, which due to their applications are of great importance, have been the starting point of a considerable literature in Approximation Theory, and in this paper we study some of the developments that have taken place around these inequalities. The paper is expository in nature and would provide results dealing with extensions, generalizations and refinements of these inequalities starting from the beginning of this subject to some of the recent ones.
$$\displaystyle{ M(p,R) \leq \Big (\frac{R^{n} + 1} {2} \Big)\vert \vert p\vert \vert,\ \ \mathrm{for}\ \ R \geq 1, }$$
(1)
$$\displaystyle{ M(p,r) \geq \Big (\frac{r + 1} {2} \Big)^{n}\vert \vert p\vert \vert,\ \ \mathrm{for}\ \ 0 <r \leq 1, }$$
(2)