The Lebesgue measure, λ (E + F), of the algebraic sum of two Borel sets, E, F of the classical “middle-thirds’ Cantor set on the circle can be estimated by evaluating the Cantor meaure, μ of the summands. For example log λ (E + F) exceeds a fixed scalar multiple of log μ (E)+ log μ (F). Several numerical inequalities which are required to prove this and related results look tantalizingly simple and basic. Here we isolate them from the measure theory and present a common format and proof.

Let f be a continuous function, and u a continuous linear function, from a Banach space into an ordered Banach space, such that f − u satisfies a Lipschitz condition and u satisfies an inequality implicit-function condition. Then f also satisfles an inequality implicit-function condition. This extends some results of Flett, Craven and S. M. Robinson.

The main aim of this note is the proof of the following

Let −∞ ≤ a > b ≤ ∞ and let A ⊂ (a, b) be a measurable set such that λ((a, b)\A) = 0, where λ denotes Lebesgue measure on ℝ. Let f: A→ℝ be a measurable and midconvex function, i.e.

whenever. Then there exists a convex functionsuch that.

On compact oriented differentiable manifolds, we define a well behaved Riemann type integral which coincides with the Lebesgue integral on nonnegative functions, and such that the exterior derivative of a differentiable (not necessarily continuously) exterior form is always integrable and the Stokes formula holds.

We study the existence, uniqueness and regularity of solutions to an exterior elliptic free boundary problem. The solutions model stationary solutions to nonlinear diffusion reaction problems, that is, they have compact support and satisfy both homogeneous Dirichlet and Neumann-type boundary conditions on the free boundary ∂{u > 0}. Then we prove convexity and symmetry properties of the free boundary and of the level sets {u > c} of the solutions. We also establish symmetry properties for the corresponding interior free boundary problem.

We say that a function has locally small Riemann sums on an interval if for each point x in the interval, and for each positive number ε, all sufficiently fine partitions of intervals lying in neighborhood of x but not containing x have Riemann sums of absolute value less than ε. The main result is then as the title states. We use the generalized Riemann approach to Perron integration, assuming that functions are measurable only to insure that conditions involving the positive and negative parts of the functions are satisfied.

We present a systematic and self-contained exposition of the generalized Riemann integral in a locally compact Hausdorff space, and we show that it is equivalent to the Perron and variational integrals. We also give a necessary and sufficient condition for its equivalence to the Lebesgue integral with respect to a suitably chosen measure.

We obtain various refinements and generalizations of a classical inequality of S. N. Bernstein on trigonometric polynomials. Some of the results take into account the size of one or more of the coefficients of the trigonometric polynomial in question. The results are obtained using interpolation formulas.

A maximality principle on quasi-ordered pseudo-metric spaces is used to obtain a number of Lipschitz attraction results for non-semigroup evolution processes with respect to time-dependent families. As particular cases, a multivalued version of Dieudonné's means value theorem and the Kirk-Ray lipschitzianness test are derived.

The special Denjoy-Bochner integral (the D*B-integral) which are generalisations of Lebesgue-Bochner integral are discussed in [7, 6, 5]. Just as the concept of numerical almost periodicity was extended by Burkill [3] to numerically valued D*- or D-integrable function, we extend the concept of almost periodicity for Banach valued function to Banach valued D*B-integrable function. For this purpose we introduce as in [3] a distance in the space of all D*B-integrable functions with respect to which the D*B-almost periodicity is defined. It is shown that the D*B-almost periodicity shares many of the known properties of the almost periodic Banach valued function [1, 4].

We introduce the notion of functions of bounded proximal variation and the notion of orderly connected topology on the real line. Using these notions, we define in a novel way an integral of Perron type, including virtually all the known integrals of Perron and Denjoy types and admitting mean value theorems and integration by parts and the analog of Marcinkiewicz theorem for the ordinary Perron integral.

If two functions of a real variable are integrable over two intervals, say of t, τ, respectively, then the product of the two functions should be integrable over the rectangular product of the two intervals of t and τ. For the Lebesgue integral, definable using non-negative functions alone, the proof is easy. For non-absolute integrals such as the Perron, Çesàro-Perron, and Marcinkiewicz-Zygmund integrals we have difficulties since the functions cannot be assumed non-negative. But the present paper gives a proof.

This paper defines descriptive, Riemann, and constructive integrals equivalent to the approximately continuous integral of Burkill.

Two index laws for fractional integrals and derivatives, which have been extensively studied by E. R. Love, are shown to be special cases of an index law for general powers of certain differential operators, by means of the theory developed in a previous paper. Discussion of the two index laws, which are rather different in appearance, can thus be unified.

The purpose of this paper is to prove the inequality of Theorem 1. The problem is due to B. Korenblum, who asked about it in connection with the characterization of the zero sets of functions analytic in the unit disc satisfying a growth condition. The problem was communicated to us by W. K. Hayman.

Under the standard assumptions, the formula for integration by parts is obtained by a straightforward calculation.

We derive some specific inequalities involving absolutely continuous functions and relate them to a norm inequality arising from Banach algebras of functions having bounded k th variation.

A classical theorem states that any open set on the real line is a countable union of disjoint open invervals. Here the numerical content of this theorem is investigated with the methods of constructive topology.

Given a simplex S and a positive function δ on S, we show that there is a simplicial subdivision of S such that the diameter of each subdividing simplex is smaller that δ evaluated at some of its vertices.

In recent papers, Russell introduced the notions of functions of bounded kth variation (BVk functions) and the RSk integral. Das and Lahiri enriched Russell's works along with a convergence formula of RSk integrals depending on the convergence of integrands. In this paper a convergence theorem analogous to Arzela's dominated convergence theorem has been presented. An investigation to the convergence in kth variation has been made leading to some convergence theorems of RSk integrals depending on the convergence of integrators.