11 results
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2. A generalization of Bohr–Mollerup's theorem for higher order convex functions: a tutorial.
- Author
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Marichal, Jean-Luc and Zenaïdi, Naïm
- Subjects
- *
CONVEX functions , *GAMMA functions , *DIFFERENCE operators , *GENERALIZATION , *FUNCTIONAL equations , *OPEN access publishing - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
3. Singular Modifications Of A Classical Function.
- Author
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Serbenyuk, S.
- Subjects
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FUNCTIONAL equations , *GENERALIZATION - Abstract
The present article deals with properties of one class of functions with complicated local structure. These functions can be modeled by certain operators of digits. Such operators were considered by the author earlier (for example, see [27,39] and references therein). This research is a generalization of investigations presented in the last-mentioned papers. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
4. The solvability of f(p(x))=q(f(x)) for given strictly monotonous continuous real functions p, q.
- Author
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Kopeček, Oldřich
- Subjects
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CONTINUOUS functions , *CHARACTERISTIC functions , *PROBLEM solving , *MATHEMATICS - Abstract
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]
- Published
- 2022
- Full Text
- View/download PDF
5. On the invariance of the arithmetic mean with respect to generalized Bajraktarević means.
- Author
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Grünwald, R. and Páles, Zs.
- Subjects
- *
ARITHMETIC mean , *LINEAR differential equations , *QUADRATIC equations , *FUNCTIONAL equations - Abstract
The purpose of this paper is to investigate the following invariance equation involving two 2-variable generalized Bajraktarević means, i.e., we aim to solve the functional equation f - 1 ( p 1 (x) f (x) + p 2 (y) f (y) p 1 (x) + p 2 (y) ) + g - 1 ( q 1 (x) g (x) + q 2 (y) g (y) q 1 (x) + q 2 (y) ) = x + y (x , y ∈ I) , where I is a nonempty open real interval and f , g : I → R are continuous, strictly monotone and p 1 , p 2 , q 1 , q 2 : I → R + are unknown functions. The main result of the paper shows that, assuming four times continuous differentiability of f, g, twice continuous differentiability of p1 and p2 and assuming that p1 differs from p2 on a dense subset of I, a necessary and sufficient condition for the equality above is that the unknown functions are of the form f = u v , g = w z , and p 1 q 1 = p 2 q 2 = v z , where u , v , w , z : I → R are arbitrary solutions of the second-order linear differential equation F ′ ′ = γ F (γ ∈ R is arbitrarily fixed) such that v > 0 and z > 0 holds on I and { u , v } and { w , z } are linearly independent. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
6. On the equality problem of two-variable Bajraktarević means under first-order differentiability assumptions.
- Author
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Páles, Zsolt and Zakaria, Amr
- Subjects
- *
FUNCTIONAL equations , *CONTINUOUS functions , *HAMMERSTEIN equations - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
7. On a new class of functional equations satisfied by polynomial functions.
- Author
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Nadhomi, Timothy, Okeke, Chisom Prince, Sablik, Maciej, and Szostok, Tomasz
- Subjects
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POLYNOMIALS , *LINEAR equations , *FUNCTIONAL equations , *MATHEMATICS , *EQUATIONS - Abstract
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]
- Published
- 2021
- Full Text
- View/download PDF
8. Conditional distributivity for semi-t-operators over uninorms.
- Author
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Wang, Wei and Qin, Feng
- Subjects
- *
UTILITY theory , *CONDITIONAL expectations , *GENERALIZATION , *AGGREGATION operators - Abstract
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]
- Published
- 2022
- Full Text
- View/download PDF
9. Complementary means with respect to a nonsymmetric invariant mean.
- Author
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Matkowski, Janusz
- Subjects
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FUNCTIONAL equations , *MATHEMATICS - Abstract
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]
- Published
- 2022
- Full Text
- View/download PDF
10. On the equality of two-variable general functional means.
- Author
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Losonczi, László, Páles, Zsolt, and Zakaria, Amr
- Subjects
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BOREL subsets , *ORDINARY differential equations , *PROBABILITY measures , *BOREL sets , *FUNCTIONAL equations - Abstract
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]
- Published
- 2021
- Full Text
- View/download PDF
11. Non-existence of measurable solutions of certain functional equations via probabilistic approaches.
- Author
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Okamura, Kazuki
- Subjects
- *
DISTRIBUTION (Probability theory) , *EQUATIONS - Abstract
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]
- Published
- 2021
- Full Text
- View/download PDF
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