A theorem on maximum modulus

A Strong Maximum Modulus Theorem for Maximal Function Algebras

On the Maximum Modulus of Polynomials. Ii

Let f(z) := ∑n ν=0 aνz ν be a polynomial of degree n having no zeros in the open unit disc, and suppose that max|z|=1 |f(z)| = 1. How small can max|z|=ρ |f(z)| be for any ρ ∈ [0 , 1)? This problem was considered and solved by Rivlin [4]. There are reasons to consider the same problem under the additional assumption that f ′(0) = 0. This was initiated by Govil [2] and followed up by the present ...

On the Structure of Maximum Modulus Algebras

i.e., if every fER attains its maximum modulus on C. As a matter of convenience, we shall, in this paper, abbreviate "maximum modulus algebra on K" to "ilf-algebra." Examples of Af-algebras which come to mind immediately are (a) the algebra ft consisting of all functions which are continuous on K and analytic in U, (b) any subalgebra of ft, (c) any algebra (B which is equivalent to an algebra o...

On the maximum modulus of a polynomial and its derivatives

Let f (z) be an arbitrary entire function and M(f ,r) = max |z|=r | f (z)|. For a polynomial P(z) of degree n, having no zeros in |z| < k, k ≥ 1, Bidkham and Dewan (1992) proved max |z|=r |P (z)| ≤ (n(r + k) n−1 /(1 + k) n)max |z|=1 |P(z)| for 1 ≤ r ≤ k. In this paper, we generalize as well as improve upon the above inequality.

The maximum modulus of a trigonometric trinomial

Let Λ be a set of three integers and let CΛ be the space of 2π-periodic functions with spectrum in Λ endowed with the maximum modulus norm. We isolate the maximum modulus points x of trigonometric trinomials T ∈ CΛ and prove that x is unique unless |T | has an axis of symmetry. This permits to compute the exposed and the extreme points of the unit ball of CΛ, to describe how the maximum modulus...