The purpose of this work is to introduce some new results about the relations between powers, roots and Moore–Penrose inverses of square matrices satisfying a cubic matrix equation and the generalized Fibonacci numbers. The results can also be used for rectangular matrices. Moreover, we give some numerical examples to verify theoretical results.
In this paper, we deal with the question: under what conditions n distinct points Pi(xi,yi) (i=1,...,n) provided x1<...<xn form a convex polygon? One of the main findings of the paper can be stated as follows: Let P1(x1,y1),..., Pn(xn,yn) be n distinct points (n≥3) with x1<...<xn. Then \overline{P1P2},...,\overline{PnP1} form a convex n-gon lying in the half-space \underline{H} = {(x,y)| x∈ℝ and y≤y1 + [(x-x1)/(xn-x1)](yn-y1)} ⊆ ℝ2 if and only if the following inequality holds (yi-yi-1)/(xi-xi-1) ≤ (yi+1-yi)/(xi+1-xi) for all i∈{2,...,n−1}. Based on this result, we establish a connection between the property of sequential convexity and convex polygon. We show that in a plane if any n points are scattered in such a way that their horizontal and vertical distances preserve some specific monotonic properties, then those points form a 2-dimensional convex polytope.
We introduce the generalized notion of semicontinuity of a function defined on a topological space and derive the useful classification of the socalled Lipschitz derivatives of functions defined on a metric space. Secondly, we investigate some connections of the Lipschitz derivatives defined on normed spaces to the Fréchet derivative and relations between little, big and local Lipschitz derivatives (denoted by lip f, Lip f and Lip f respectively) in terms of Baire limit functions. In particular, we prove that lip f is Fσ-lower, Lip f is Fσ-upper, Lip f is upper semicontinuous. Moreover, for a function f defined on an open or convex subset of a normed space, the upper Baire limit function of functions lip f and Lip f are equal to Lip f.
Let S be a semigroup and K be a field. In a recent article we introduced a new cosine functional equation g(xyz)−g(x)g(yz)−g(y)g(xz)−g(z)g(xy)+2g(x)g(y)g(z)=0 for an unknown function g:S→K. It was shown that this equation is closely connected to the sine addition formula, and for K=C its solutions are expressible in terms of multiplicative functions. Here we solve the more general functional equation f(xyz)+g(x)g(yz)+g(y)g(xz)+g(z)g(xy)+h(x)h(y)h(z)=0 for three unknown functions f,g,h:S→C, where S is a monoid. The solutions are linear combinations of two multiplicative functions.
In this paper, we use (m+4)-convex functions to derive an estimate for Jensen’s inequality in the context of divided differences. In addition, we extend these results for (h,g;α−n)-convex functions. Finally, we present some results for g-convex functions, (h,g)-convex functions and provide a discussion and examples concerning h-convex functions.
10.2478/amsil
English