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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]

We investigate the functional equation f (p (x)) = q (f (x)) where p and q are given real functions. In the paper "On solvability of f (p (x)) = q (f (x)) for given real functionsp, q, Aequat. Math. 90 (2016), 471 - 494", we solved the problem of the solvability of f (p (x)) = q (f (x)) under the assumption that p, q are strictly increasing continuous real functions. Now, we extend the solutions of this problem for any strictly monotonous continuous real functions p, q. Thereby, we use the methods of the just mentioned paper. Further, we present computations of the so called characteristics of the given functions p, q using the results of this paper and, finally, present a quite short algorithm with input p, q and output 'solvable/not solvable'. [ABSTRACT FROM AUTHOR]

The equality problem of two-variable Bajraktarević means can be expressed as the functional equation f g - 1 ( f (x) + f (y) g (x) + g (y) ) = h k - 1 ( h (x) + h (y) k (x) + k (y) ) (x , y ∈ I) , where I is a nonempty open real interval, f , g , h , k : I → R are continuous functions, g, k are positive and f/g, h/k are strictly monotone. This functional equation, for the first time, was solved by Losonczi in 1999 under 6th-order continuous differentiability assumptions. Additional and new characterizations of this equality problem have been found recently by Losonczi, Páles and Zakaria under the same regularity assumptions in 2021. In this paper it is shown that the same conclusion can be obtained under substantially weaker regularity conditions, namely, assuming only first-order differentiability. [ABSTRACT FROM AUTHOR]

The classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi's result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation F (x + y) - F (x) - F (y) = y f (x) + x f (y) considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation. [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 known that if a bivariate mean K is symmetric, continuous and strictly increasing in each variable, then for every mean M there is a unique mean N such that K is invariant with respect to the mean-type mapping M , N , which means that K ∘ M , N = K and N is called a K-complementary mean for M (Matkowski in Aequ Math 57(1):87–107, 1999). This paper extends this result for a large class of nonsymmetric means. As an application, the limits of the sequences of iterates of the related mean-type mappings are determined, which allows us to find all continuous solutions of some functional equations. [ABSTRACT FROM AUTHOR]

Given two functions f , g : I → R and a probability measure μ on the Borel subsets of [0, 1], the two-variable mean M f , g ; μ : I 2 → I is defined by M f , g ; μ (x , y) : = (f g ) - 1 ∫ 0 1 f (t x + (1 - t) y) d μ (t) ∫ 0 1 g (t x + (1 - t) y) d μ (t) (x , y ∈ I). This class of means includes quasiarithmetic as well as Cauchy and Bajraktarević means. The aim of this paper is, for a fixed probability measure μ , to study their equality problem, i.e., to characterize those pairs of functions (f, g) and (F, G) for which M f , g ; μ (x , y) = M F , G ; μ (x , y) (x , y ∈ I) holds. Under at most sixth-order differentiability assumptions for the unknown functions f, g and F, G, we obtain several necessary conditions in terms of ordinary differential equations for the solutions of the above equation. For two particular measures, a complete description is obtained. These latter results offer eight equivalent conditions for the equality of Bajraktarević means and of Cauchy means. [ABSTRACT FROM AUTHOR]

This paper deals with functional equations in the form of f (x) + g (y) = h (x , y) where h is given and f and g are unknown. We will show that if h is a Borel measurable function associated with characterizations of the uniform or Cauchy distributions, then there is no measurable solutions of the equation. Our proof uses a characterization of the Dirac measure and it is also applicable to the arctan equation. [ABSTRACT FROM AUTHOR]