On Bounds of Polynomials in Hyperspheres and Frechet-Michal Derivatives in Real and Complex Normed Linear Spaces

Abstract

Introduction: One of the main difficulties-by no means the only onein the development of a theory of analytic functions in normed linear spaces stems from the fact that the modulus of a homogeneous polynomial and that of its polar are not always the same. We give here the theorenm that asserts that the modulus of a homogeneous polynomial and its polar are always equal in complex' Banach spaces. This has many important imi lications including the great simplifications brought about in the proofs of several furndamental theorems. The situation in real2 Banach spaces is quite different as we shall show. We base our proofs on a few fundamental lemmas in normed linear spaces: one generalizes Bernstein's theorem3 on the bounds and first derivative of S, a polynomial in the complex plane and two others generalize theorems of the brothers, A. Markoff4 and V. Markoff5 on the first and higher derivatives of polynomials of a real variable. These lemmas have an interest in themselves and have many applications not discussed in this paper. To present our results as briefly as possible, we assume familiarity with the Frechet differential calculus and analytic function theory in both real and complex Banach spaces. Theorems in Real Banach Spaces. The result of A. Markoff states that if pn(x) is a polynomial of degree n in a real variable x and if lPn(x)I < 1 in the interval xI < 1, then the derivative pn(x) satisfies the inequality lpA(x)l < nZ in |x| < 1. The following lelmma generalizes Markoff's result. Lemma 1. Let n