Improvement and generalization of polynomial inequality of T. J. Rivlin

Relative Growth of a Complex polynomial with Restricted Zeros

“…In this paper, we continue the study of this type of inequality for polynomials, following up on a study started by various authors in the recent past. Generally, under similar hypotheses, Rivlin's inequality has been improved and generalized by adopting two different approaches (see the papers [13,7,4,5,10,3,11]) to mention only a few. More precisely, we adopt one of these approaches and established some new inequalities while taking into account the placement of the zeros of the underlying polynomial.…”

“…We begin with the well-known Bernstein inequalities [2] for the uniform norm on the unit disk in the plane: namely, if p(z) is a polynomial of degree n, then Inequality (1.1) is a direct consequence of Bernstein's theorem on the derivative of a trigonometric polynomial [16], and inequality (1.2) follows from the maximum modulus theorem (see [14,Problem 269]). The reverse analogue of the inequality (1.2) whenever R ≤ 1 is given by Varga [17], and he proved that, if p(z) is a polynomial of degree n, then for 0 ≤ r ≤ 1 For the class of polynomials having no zero inside the unit circle, it was Rivlin [15] who proved that, if p(z) is a polynomial of degree n having no zero in |z| < 1, then for 0 ≤ r ≤ 1 The above inequalities are the starting point of a rich literature concerning their extensions, generalizations and improvements in several directions, see the papers ( [1,9,13,7,4,5,10,3,11]) to mention only a few. For a deeper understanding about this kind of inequalities and their applications, we refer to the monographs [12,8].…”