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In this paper, we utilize some known Steffensen-type identities, obtained by using Lidstone's interpolating polynomial, to prove new generalizations of Steffensen's inequality. We obtain these new generalizations by using the weighted Hermite-Hadamard inequality for (2 n + 2) - convex and (2 n + 3) - convex functions. Further, the newly obtained inequalities can be observed as an upper- and lower-bound for utilized Steffensen-type identities. [ABSTRACT FROM AUTHOR]

In its additive version, Bohr–Mollerup's remarkable theorem states that the unique (up to an additive constant) convex solution f(x) to the equation Δ f (x) = ln x on the open half-line (0 , ∞) is the log-gamma function f (x) = ln Γ (x) , where Δ denotes the classical difference operator and Γ (x) denotes the Euler gamma function. In a recently published open access book, the authors provided and illustrated a far-reaching generalization of Bohr–Mollerup's theorem by considering the functional equation Δ f (x) = g (x) , where g can be chosen from a wide and rich class of functions that have convexity or concavity properties of any order. They also showed that the solutions f(x) arising from this generalization satisfy counterparts of many properties of the log-gamma function (or equivalently, the gamma function), including analogues of Bohr–Mollerup's theorem itself, Burnside's formula, Euler's infinite product, Euler's reflection formula, Gauss' limit, Gauss' multiplication formula, Gautschi's inequality, Legendre's duplication formula, Raabe's formula, Stirling's formula, Wallis's product formula, Weierstrass' infinite product, and Wendel's inequality for the gamma function. In this paper, we review the main results of this new and intriguing theory and provide an illustrative application. [ABSTRACT FROM AUTHOR]

In this research paper, we investigate a generalization of Vincze's type functional equations involving several (up to four) unknown functions in connection with the maximum functional equation as max { ψ (x y) , ψ (x y - 1) } = ψ (x) η (y) + ψ (y) , max { ψ (x y) , ψ (x y - 1) } = ψ (x) η (y) + χ (y) , max { ψ (x y) , ψ (x y - 1) } = ϕ (x) η (y) , max { ψ (x y) , ψ (x y - 1) } = ϕ (x) η (y) + χ (y) , where G is an arbitrary group, x , y ∈ G , and ψ , η , χ , ϕ : G → R are unknown functions. [ABSTRACT FROM AUTHOR]

In this paper we will study a connection between some generalization of AC-subspaces, vector valued invariant λ -means and λ -complementability. [ABSTRACT FROM AUTHOR]

Let G be a graph with vertex set V(G) and edge set E(G), and let f be an integer-valued function defined on V(G). It is proved in this paper that every bipartite (0 , m f - m + 1) -graph has a (0, f)-factorization randomly r-orthogonal to n vertex-disjoint mr-subgraphs of G, which is a generalization of the known result with n = 1 given by Zhou and Wu. [ABSTRACT FROM AUTHOR]

Being a universal space for weight A ≥ ℵ 0 metric spaces Lipscomb's space J A has a central role in topological dimension theory. There exists a strong connection between topological dimension theory and fractal set theory since on the one hand, some classical fractals play the role of universal spaces and on the other hand the universal space J A is a generalized Hutchinson–Barnsley fractal (i.e. the attractor of a possibly infinite iterated function system). In this paper we introduce a generalization of J A , namely the concept of graph Lipscomb's space J A G associated with a graph G on the set A, and we prove that its imbedded version in l 2 (A ′) , where A ′ = A \ { z } , z being a fixed element of the set A having at least two elements, is a generalized Hutchinson–Barnsley fractal. [ABSTRACT FROM AUTHOR]

For a given p-variable mean M:Ip→I\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M :I^p \rightarrow I$$\end{document} (I is a subinterval of R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}$$\end{document}), following (Horwitz in J Math Anal Appl 270(2):499–518, 2002) and (Lawson and Lim in Colloq Math 113(2):191–221, 2008), we can define (under certain assumptions) its (p+1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(p+1)$$\end{document}-variable β\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\beta $$\end{document}-invariant extension as the unique solution K:Ip+1→I\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K :I^{p+1} \rightarrow I$$\end{document} of the functional equation K(M(x2,⋯,xp+1),M(x1,x3,⋯,xp+1),⋯,M(x1,⋯,xp))=K(x1,⋯,xp+1),for allx1,⋯,xp+1∈I\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned}&K\big (M(x_2,\dots ,x_{p+1}),M(x_1,x_3,\dots ,x_{p+1}),\dots ,M(x_1,\dots ,x_p)\big )\\&\quad =K(x_1,\dots ,x_{p+1}), \text { for all }x_1,\dots ,x_{p+1} \in I \end{aligned}$$\end{document}in the family of means. Applying this procedure iteratively we can obtain a mean which is defined for vectors of arbitrary lengths starting from the bivariate one. The aim of this paper is to study the properties of such extensions. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new class of dynamic inequalities of Hardy's type which generalize and improve some recent results given in the literature. More precisely, we prove some new Hardy-type inequalities involving many functions on time scales. Some new discrete inequalities are deduced in seeking applications. [ABSTRACT FROM AUTHOR]

In this paper, the author discusses a generalization of Schröder's and Abel's functional equation of the form ∑ n α n (x) f (u n (x)) = g (x). In fact, using the theory of operators and infinite matrices, we show under certain conditions this equation has only a unique bounded solution f : X → R. [ABSTRACT FROM AUTHOR]

The conditional distributivity between two different aggregation operators, which has received wide attention from the researchers, is vital for many fields, for example, utility theory, integration theory and so on. In some existing generalization, the restrictive but not completely justified condition that the values of the inner operator are less than 1. However, for a more general and reasonable setting, the values of the inner operator should be strictly bounded between 0 and 1. Therefore, the aim of this paper is to introduce and fully characterize this kind of conditional distributivity of a semi-t-operator over a uninorm. In comparison with the corresponding results obtained, there are many new solutions. [ABSTRACT FROM AUTHOR]

It is well known that Heron's equality provides an explicit formula for the area of a triangle, as a symmetric function of the lengths of its edges. It has been extended by Brahmagupta to quadrilaterals inscribed in a circle (cyclic quadrilaterals). A natural problem is trying to further generalize the result to cyclic polygons with a larger number of edges. Surprisingly, this has proved to be far from simple, and no explicit solutions exist for cyclic polygons having n > 4 edges. In this paper we investigate such a problem by following a new and elementary approach, based on the idea that the simple geometry underlying Heron's and Brahmagupta's equalities hides the real players of the game. In details, we propose to focus on the dissection of the edges determined by the incircles of a suitable triangulation of the cyclic polygon, showing that this approach leads to an explicit formula for the area as a symmetric function of the lengths of these segments. We also show that such a symmetry can be rediscovered in Heron's and Brahmagupta's results, which consequently represent special cases of the provided general equality. [ABSTRACT FROM AUTHOR]

The Gárfás–Sumner conjecture asks whether for every tree T, the class of (induced) T-free graphs is χ -bounded. The conjecture is solved for several special trees, but it is still open in general. Motivated by the conjecture, the chromatic number of triangle-free and broom-free graphs is well studied, since a broom is one of the generalizations of a star, where a broomB(m, n) is the graph obtained from a star K 1 , n and an m-vertex path P m by identifying the center of K 1 , n and a leaf of P m . Gárfás, Szemeredi and Tuza proved that for every triangle-free and B(m, n)-free graph G, χ (G) ≤ m + n - 1 . This upper bound has been improved by Wang and Wu to m + n - 2 for m ≥ 2 , n ≥ 1 . In this paper, we prove that any triangle-free and B(4, 2)-free graph G is 3-colorable if the number of vertices of G is at least 17. Furthermore, the above estimation is the best possible since there exists a triangle-free and B(4, 2)-free 4-chromatic graph with sixteen vertices, named the Clebsch graph. [ABSTRACT FROM AUTHOR]

In this paper, we establish some conditions on nonnegative rd-continuous weight functions u x and υ x which ensure that a reverse dynamic inequality of the form ∫ a ∞ f p (x) υ x Δ x 1 p ≤ C ∫ a ∞ u x ∫ a σ x K σ x , σ y f (y) Δ y q Δ x 1 q , holds when q ≤ p < 0 and 0 < q ≤ p < 1. Corresponding dual results are also obtained. In particular, we prove some reverse dynamic weighted Hardy-type inequalities with kernels on time scales which as special cases contain some generalizations of integral and discrete inequalities due to Beesack and Heinig. [ABSTRACT FROM AUTHOR]

In (Brzdęk and Schwaiger in Aeq Math 92: 975–991, 2018) solutions of far reaching generalizations of the so-called radical functional equation f (p (π (x) + π (y))) = f (x) + f (y) have been investigated. These investigations are continued here by analysing the corresponding stability results, which have been the main subject of several recent papers. We propose a very general and uniform approach. [ABSTRACT FROM AUTHOR]

In Borowiecka et al. (Bernoulli 21(4):2513-2551, 2015) the authors show that every generalized convolution can be used to define a Markov process, which can be treated as a Lévy process in the sense of this convolution. The Bessel process is the best known example here. In this paper we present new classes of regular generalized convolutions enlarging the class of such Markov processes. We give here a full characterization of such generalized convolutions ⋄ for which δx⋄δ1, x∈[0,1], is a convex linear combination of n=3 fixed measures and only the coefficients of the linear combination depend on x. For n=2 it was shown in Jasiulis-Goldyn and Misiewicz (J Theor Probab 24(3):746-755, 2011) that such a convolution is unique (up to the scale and power parameters). We show also that characterizing such convolutions for n⩾3 is equivalent to solving the Levi-Civita functional equation in the class of continuous generalized characteristic functions. [ABSTRACT FROM AUTHOR]

This paper presents a mathematical generalization of Halphén's formula for derivatives. It notes that since the identity is of an algebraic nature among derivatives, it automatically extend to non-analytic functions of sufficient smoothness by a general principle. Mathematical equations that proves the main formula is offered.

The article presents a paper that establishes inequalities of Hammer-Bullen type for convex sequences and nondecreasing convex function. The inequalities were established through the use of Wu-Debnath's method. The inequality of Fejér-Hammer-Bullen type for convex sequences was derived by assuming symmetry of an involved sequence.

This paper studies the solutions of the integral Van Vleck's functional equation for the sine and of the integral Kannappan's functional equation. Topics discussed include complex-valued solutions of the functional equation, d'Alembert's functional equation, and generalization of the lemmas of H. Stetkær.

We consider cotangent sums associated to the zeros of the Estermann zeta function considered by the authors in their previous paper (Maier and Rassias, Generalizations of a cotangent sum associated to the Estermann zeta function, ). We settle a question on the rate of growth of the moments of these cotangent sums left open in Maier and Rassias (Generalizations of a cotangent sum associated to the Estermann zeta function, ), and obtain a simpler proof of the equidistribution of these sums. [ABSTRACT FROM AUTHOR]

The structure of disjoint iteration groups of real homeomorphisms has been determined by M. C. Zdun without any regularity condition. In this paper we turn to the regular case and describe the structure of regular disjoint groups of real homeomorphisms which are generalizations of regular disjoint iteration groups. It is shown that such a group is either embedded in a regular iteration group, or it is homeomorphically conjugate to a regular piecewise linear disjoint group. [ABSTRACT FROM AUTHOR]

The main objective of this article is to introduce a new class of real valued functions that include the well-known class of $${m}$$ -convex functions introduced by Toader (). The members of this collection are called Jensen $${m}$$ -convex and are defined, for $${m \in (0,1]}$$ , via the functional inequality where $${c_{m} := \frac{m+1}{m}}$$ . These functions generate a new kind of functional convexity that is studied in terms of its behavior with respect to basic algebraic operations such as sums, products, compositions, etc. in this paper. In particular, it is proved that any starshaped Jensen convex function is Jensen $${m}$$ -convex. At the same time an interesting example (Example 3) shows how the classes of Jensen $${m}$$ -convex functions depend on $${m}$$ . All the techniques employed come from traditional basic calculus and most of the calculations have been done with Mathematica 8.0.0 and validated with Maple 15 as well as all the figures included. [ABSTRACT FROM AUTHOR]