«modulus of continuity»

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 pro

Correction and Extension of van der Poel/s Method for Calculating the Shear Modulus of a Particulate Composite

Van de r Poel's method (Rheol. Ac ta 1, 198 (1958)) for ca lculating the shear modu lus of a par· ti culate composite agrees well with experimental data, but its validity has been ques tioned, and it was applicable only to composites in whi ch the matrix materia l is incompress i

Sharp Estimates for the Maximum over Minimum Modulus of Rational Functions

Let m, n ≥ 0, λ > 1, and R be a rational function with numerator, denominator of degree ≤ m,n, respectively. In several applications, one needs to know the size of the set S ⊂ [0, 1] such that for r ∈ S, max |z|=r |R (z)| / min |z|=r |R (z)| ≤ λ. In an earlier paper, we showed th

The Minimum Modulus of Large Integral Functions

Introduction 1. THIS paper embodies mainly the proof of results announced previously (Hayman (5)). Let f(z) be a non-constant integral function and p u t

Normal Structure and Modulus of U -convexity in Banach Spaces

Let X be a normed linear space, and let S(X) = fx 2 X : kxk = 1g be the unit sphere of X. Brodskii and Milman 1] introduced the following geometric concept: Deenition 1. A bounded, convex subset K of a Banach space X is said to have normal structure if every convex subset H of K

On the Modulus of U-convexity

We prove that the moduli of U-convexity, introduced by Gao (1995), of the ultrapower X̃ of a Banach space X and of X itself coincide whenever X is super-reflexive. As a consequence, some known results have been proved and improved. More precisely, we prove that uX(1) > 0 implies

Inequalities for the Maximum Modulus of the Derivative of a Polynomial

Let P (z) be a polynomial of degree n and M(P, t) = Max|z|=t |P (z)|. In this paper we shall estimate M(P ′ , ρ) in terms of M(P, r) where P (z) does not vanish in the disk |z| ≤ K, K ≥ 1, 0 ≤ r < ρ < K and obtain an interesting refinement of some result of Dewan and Malik. We sh

Weighted Maximum over Minimum Modulus of Polynomials, Applied to Ray Sequences of Padé Approximants

Let a 0; " > 0. We use potential theory to obtain a sharp lower bound for the linear Lebesgue measure of the set ( r 2 [0; 1] : r maxjtj=1 jP (t)j minjtj=r jP (t)j " ) : Here P is an arbitrary polynomial of degree n. We then apply this to diagonal and ray Padé sequences for funct

On the Ratio of the Modulus of Dirichlet Eta Function Values of Critical Line Symmetrical Arguments and the Riemann Hypothesis

ABSTRACT. Said Sn(s) the nth partial sum of the Dirichlet η function, Rn(s) the corresponding remainder term, and ρ = 2 +α + it , τ = 1 2 −α + it , a pair of arguments symmetrical with respect to the critical line, and restricting the following considerations to the domain corres

Linear Modulus of a Linear Operator