Let Λ ⊆ {1, . . . , N}, and let {an}n∈Λ be a sequence with |an| ≤ 1 for all n. It is easy to see that ∥∥∥∥ ∑ n∈Λ ane(nθ) ∥∥∥∥ p ≤ ∥∥∥∥ ∑ n∈Λ e(nθ) ∥∥∥∥ p for every even integer p. We give an example which shows that this statement can fail rather dramatically when p is not an even integer. This answers in the negative a question known as the Hardy-Littlewood majorant conjecture, thereby ruling ...

This paper uses majorant techniques to study the convergence of Broyden's single-rank update method for nonlinear systems of equations. It also contains a very elementary proof of the local convergence of the method. The heart of the method is a procedure for generating an approximation to the Jacobian of the system using only information on hand and not requiring partial derivatives.

Using potential theoretic methods we give extension results for functions in various Hardy classes and in the Nevanlinna class in C". Our exceptional sets are polar or slightly larger n-small sets depending whether the majorant is a harmonic or separately hyperharmonic function.

We present a logarithmic barrier interior-point method for solving a second-order cone programming problem. Newton’s method is used to compute the descent direction, and majorant functions are used as an efficient alternative to line search methods to determine the displacement step along the direction. The efficiency of our method is shown by presenting numerical experiments.

A new technique is used instead of the classical majorant principle to analyze the R-order of convergence of the Newton process when more general conditions than the Kantorovich ones are considered. © 2005 Elsevier B.V. All rights reserved. MSC: 47H17; 65J15

We study the behavior of maximal term Dirichlet series with positive exponents sum which is an entire function. For class determined by a convex growth majorant we establish criterion for equivalence logarithms moduli original and modified outside some exceptional set.

We provide a representation of the maximal difference between a standard Brownian bridge and its concave majorant on the unit interval, from which we deduce expressions for the distribution and density functions and moments of this difference. This maximal difference has an application in nonparametric statistics where it arises in testing monotonicity of a density or regression curve.

We prove that each positive operator from a Banach lattice E to a Banach lattice F with a disjointly strictly singular majorant is itself disjointly strictly singular provided the norm on F is order continuous. We prove as well that if S : E → E is dominated by a disjointly strictly singular operator, then S2 is disjointly strictly singular.