We consider a class of algebraic inequalities for functions of n variables depending on parameters that generalise the case of GA-convex functions. The functions in this class are GA-convex only in a subdomain of definition yet the inequality for GAconvexity still holds on the whole domain if suitable conditions are satisfied by the parameters. The method is elementary and allows us to give fur...

We introduce some new classes of words and permutations characterized by the second difference condition π(i − 1) + π(i + 1) − 2π(i) ≤ k, which we call the k-convexity condition. We demonstrate that for any sized alphabet and convexity parameter k, we may find a generating function which counts k-convex words of length n. We also determine a formula for the number of 0-convex words on any fixed...

1. L’Hôpital’s Rule 1 1.1. The Cauchy Mean Value Theorem 1 1.2. L’Hôpital’s Rule 2 2. Newton’s Method 5 2.1. Introducing Newton’s Method 5 2.2. A Babylonian Algorithm 6 2.3. Questioning Newton’s Method 7 2.4. Introducing Infinite Sequences 7 2.5. Contractions and Fixed Points 8 2.6. Convergence of Newton’s Method 10 2.7. Quadratic Convergence of Newton’s Method 11 2.8. An example of nonconverge...

The space of entire functions represented by Dirichlet series of several complex variables has been studied by S. Dauod [1]. M.D. Patwardhan [6] studied the bornological properties of the space of entire functions represented by power series. In this work we study the bornological aspect of the space Γ of entire functions represented by Dirichlet series of several complex variables. By Γ we den...

We introduce a finite element construction for use on the class of convex, planar polygons and show it obtains a quadratic error convergence estimate. On a convex n-gon, our construction produces 2n basis functions, associated in a Lagrange-like fashion to each vertex and each edge midpoint, by transforming and combining a set of n(n + 1)/2 basis functions known to obtain quadratic convergence....

insisting that the equality sign holds when k = n. Here, x[X] > • • • > x[n] are the xt arranged in decreasing order and, similarly, y[X] > • • • > y[n]. If (1) is only required for the increasing (decreasing) convex functions on R then one speaks of weak sub-majorization x >, respectively). The first is equivalent to (2). Let & be an open convex subset of R...

For a convex body K ⊂ Rn, the kth projection function of K assigns to any k-dimensional linear subspace of Rn the k-volume of the orthogonal projection of K to that subspace. Let K and K0 be convex bodies in Rn, let K0 be centrally symmetric and satisfy a weak regularity assumption. Let i, j ∈ N be such that 1 ≤ i < j ≤ n − 2 with (i, j) 6= (1, n−2). Assume that K and K0 have proportional ith p...

One of the classical results of the regularity theory of convex functions is the theorem of F. Bernstein and G. Doetsch [1] which states that if a real valued Jensen-convex function defined on an open interval I is bounded from above on a subinterval of I then it is continuous. According to a related result by W. Sierpiński [3], the Lebesgue measurability of a Jensen-convex function implies its...