SOME COMPACT GENERALIZATIONS OF WELL-KNOWN INEQUALITIES FOR POLYNOMIALS

Abstract

Let P(z) be a polynomial of degree n. In this paper, we consider a problem of
investigating the dependence of
| P(Rk^2 z) - \aplha P(k^2 z) + \beta { ( \frac{Rk+1}{k+1} )^n - |\alpha| } P(k^2 z) |
on maximum and minimum of |P(z)| on |z|= k for arbitrary real or complex numbers \alpha, \beta in C
with |\alha|<=1, |\beta|<=1, R > 1, k >= 1 and establish certain sharp compact generalizations of
well-known Bernstien-type inequalities for polynomials, from which a variety of interesting
results follows as special cases. Besides we shall rst obtain an interesting result which yields
a number of well-known polynomial inequalities as special cases.