Bernstein Type Inequalities Concerning Growth of Polynomials

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

$$\displaystyle{ M(p,R) \leq \Big (\frac{R^{n} + 1} {2} \Big)\vert \vert p\vert \vert,\ \ \mathrm{for}\ \ R \geq 1, }$$

(1)

and

$$\displaystyle{ M(p,r) \geq \Big (\frac{r + 1} {2} \Big)^{n}\vert \vert p\vert \vert,\ \ \mathrm{for}\ \ 0 <r \leq 1, }$$

(2)

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.