The Nineteenth Katowice Debrecen–Winter Seminar on Functional Equations and Inequalities Zakopane (Poland), January 30–February 2, 2019

The Nineteenth Katowice–Debrecen Winter Seminar on Functional Equations and Inequalities was held in Hotel Geovita in Zakopane, Poland, from January 30 to February 2, 2019. The meeting was organized by the Institute of Mathematics of the University of Silesia. 16 participants came from the University of Debrecen (Hungary), 13 from the University of Silesia in Katowice (Poland) and one from the Pedagogical University of Cracow (Poland). Professor Maciej Sablik opened the Seminar and welcomed the participants to Zakopane. The scientific talks presented at the Seminar focused on the following topics: equations in a single variable and in several variables, iteration theory, equations on abstract algebraic structures, regularity properties of the solutions of certain functional equations, functional inequalities, Hyers–Ulam stability, functional equations and inequalities involving mean values, generalized convexity. Interesting discussions were generated by the talks. There was also a Problem Session and a festive dinner. The closing address was given by Professor Zsolt Páles. His invitation to the Twentieth Debrecen–Katowice Winter Seminar on Functional Equations and Inequalities in January 2020 in Hungary was gratefully accepted. Summaries of the talks in alphabetical order of the authors follow in section 1, problems and remarks in chronological order in section 2, and the list of participants in the final section.


Abstracts of talks
Roman Badora: The Ulam stability problem for the equation f (x g(y)) = f (x)f (y) The functional equation f (x g(y)) = f (x)f (y) is a joint generalization of the exponential equation f (x + y) = f (x)f (y) and the equation f (xf (y)) = f (x)f (y). Its origin is applied to turbulent fluid motion in the averaging theory. This equation is connected with some linear operators, that is, the Reynolds operator (M.-L. Dubreil-Jacotin (1953), Y. Matras (1969)), the averaging operator, the multiplicatively symmetric operator (J. Aczél and J. Dhombres (1989) The stability problem for the equation f (x g(y)) = f (x)f (y) was considered by A. Najdecki (2007) and J. Chung (2014). We discuss the Ulam stability problem for this equation for vector-valued mappings.

Karol Baron: Weak limit of iterates of some random-valued functions and its application
Given a probability space (Ω, A, P ), a complete and separable metric space X with the σ-algebra B of all its Borel subsets, and a B ⊗ A-measurable and contractive in mean f : X × Ω → X we consider iterates (f n (x, ·)) n∈N of f , defined on X×Ω N by f 0 (x, ω) = x and f n (x, ω) = f f n−1 (x, ω), ω n for n ∈ N (cf. [4,Sec. 1.4]), and obtained in [1] the weak limit π f of this sequence. Basing on some properties of π f established in [3] and on the main result of [2], given a Lipschitz F mapping X into a separable Banach space Y we characterize solvability of the equation in the class of Lipschitz functions ϕ : X → Y by The aim of the talk is to present an elementary way to fractals which completely avoids advanced analysis and uses purely naive set theory. The method relies on fixed point methods, where the role of the Banach Contraction Principle is replaced by a slightly improved version of the Knaster-Tarski Fixed Point Theorem.

Zoltán Boros: Monomially linked multiadditive functions
Applying a classical theorem [2, Theorem 1] and following arguments similar to those in a recent paper by Masaaki Amou [1], we establish the following statement.
Theorem. If n ∈ N , 1 < k j ∈ N (j = 1, 2, . . . , n), C ∈ R , and the multiadditive function F : R n → R satisfies the functional equation for all x j ∈ R (j = 1, 2, . . . , n), then F = I∈K F I , where the function F I : R n → R is a derivation or a linear function with respect to its jth variable whenever j does or does not belong to the index set I , respectively. Here K consists of all subsets I of {1, 2, . . . , n} that satisfy the equation j∈I k j = C . It is supposed that j∈∅ k j = 1 (i.e., in case C = 1 we obtain that F is multilinear).
Pál Burai: Convexity generated by certain circulant matrices (Joint work with Judit Makó and Patrícia Szokol) In this presentation we examine the properties of Schur-convex functions, that are induced by certain circulant matrices. More precisely, let t = (t 1 , . . . , t n ) be a probability vector, and Then, a function f : Borbála Fazekas: Numerical solutions of the Lane-Emden-equation on the unit disc The Lane-Emden-equation is the following nonlinear second order partial differential equation where B ⊂ R 2 denotes the unit disc, p > 1. The solution u is in the Sobolev space H 1 0 (B). It can be proven analytically, that equation (1) has some radialsymmetric and also some radially nonsymmetric solutions. Our main task is to find these and also other solutions numerically.

György Gát: Cesàro means with varying parameters of Walsh-Fourier series
Let x be an element of the unit interval I := [0, 1). The N nth Walsh function at x ∈ I is ) introduced the notion of the Cesàro means of a Fourier series with varying parameters. That is, α = (α n ) is a sequence. Taking into account the above it is an interesting situation to investigate the behavior of σ α n n f when α n → 0. In this talk, we show some recent almost everywhere convergence results of the kind above with respect to the Walsh-Fourier series.

Roman Ger: A short proof of alienation of additivity from quadraticity
Without a use of pexiderized versions of abstract polynomials theory we show that on 2-divisible groups the functional equation forces the unknown functions f and g to be additive and quadratic, respectively, modulo a constant.
Motivated by the observation that the equation implies both the additivity and multiplicativity of f we deal also with the alienation phenomenon of equations in a single and several variables.

Attila Gilányi: Alienness of linear functional equations
The concept of the alienness of functional equations was introduced by Jean Dhombres in his paper [2]. Investigations related to it were performed by several authors during the last years (cf., e.g., the recent survey [3]).
In this talk, we present a computer assisted approach to its consideration in connection with linear functional equations. Our studies are based on a computer program developed for the solution of linear functional equations of two variables described in [1] (cf. also [4], [5] and [6]).
The main results of the paper offer various characterizations of Φ-convexity. One of the main result states that f is Φ-convex if and only if f can be decomposed into the sum of a convex function and a Φ-Hölder function.
Eszter Gselmann: On a class of linear functional equations without the range condition (Joint work with Gergely Kiss and Csaba Vincze) The main purpose of this talk is to provide the general solution for a class of linear functional equations. Let n ≥ 2 be an arbitrarily fixed integer, let further X and Y be linear spaces over the field K and let α i , β i ∈ K, i = 1, . . . , n be arbitrarily fixed constants. We describe all those functions Additionally, necessary and sufficient conditions are also given which guarantee the solutions to be non-trivial.
This equation belongs to the class of linear functional equations, that was thoroughly investigated by L. Székelyhidi in [1,2,3]. At the same time, we cannot state that the functions involved are polynomials. This is because of the fact that the homomorphisms appearing in the argument of the unknown functions do not fulfill range condition appearing in the mentioned works of L. Székelyhidi.
Tibor Kiss: Equality of Cauchy means and quasi-arithmetic means of two variables (Joint work with Zsolt Páles) The Cauchy Mean Value Theorem states that, having a nonempty open subinterval I ⊆ R and two differentiable functions f, g : I → R, for all x, y ∈ I with x = y, there exists u in the interval determined by x and y such that holds.
It is easy to check that u has to be unique provided that 0 / ∈ g (I) and f /g is invertible. In this latter case, the point u is called the Cauchy mean value of x and y, is denoted by u = C f,g (x, y), and can be expressed as The aim of the talk is to characterize those Cauchy means, for which there exists a continuous, strictly monotone function ϕ : I → R such that or, shortly, which can be written as a quasi-arithmetic mean. Only conditions required in the definitions are used.  Gergő Nagy: Characterizations of centrality in C * -algebras in terms of local monotonicity and local additivity of functions In this talk, we give characterizations of central elements in an arbitrary C * -algebra A in terms of local properties of maps on A given by the function calculus. One of these properties is local monotonicity. We say that a real function f is locally monotone at the self-adjoint element a ∈ A if for any element b ∈ A satisfying a ≤ b, one has f (a) ≤ f (b). We present a result stating that if f is defined on an open interval which is unbounded from above and it is strictly convex and increasing, then a is central if and only if f is locally monotone at a. That assertion significantly improves similar theorems by Ogasawara, Pedersen, Wu, Molnár and Virosztek. Another local property discussed in the talk is local additivity which is defined similarly as local monotonicity. A statement on local additivity analogous to the previous one is also presented as well as some applications of these results.

Andrzej Olbryś: On a functional inequality connected with the concept of T -Schur convexity (Joint work with Tomasz Szostok)
Following [2] where the concept of T -Schur convexity was introduced we examine functions f : D → R which satisfy the functional inequality of the form is a doubly stochastic matrix and D stands for a convex subset of a real linear space.
Zsolt Páles: On the homogenization of means (Joint work with Paweł Pasteczka) The aim of this talk is to introduce two notions of homogenization of means. In general, the homogenization is an operator which attaches a homogeneous mean to a given one.
To explain the main ideas, let I be an interval with inf I = 0 and let M : I n → I be an n-variable mean. We consider the following constructions: Our results show that, under some regularity or convexity assumptions, the homogenization of quasiarithmetic means are power means, and homogenization of semideviation means are homogeneous semideviation means.

Maciej Sablik: Further results on non-symmetric equations stemming from MVT
We continue our research started in [2]. We consider the equation where Φ : R 2 −→ R 2 is a piecewise linear mapping, and both f and g are unknown. The problem has arised in connection with the paper [1].

Ekaterina Shulman: Stability problems for set-valued mappings on groups
Let G be a group and let S(X) be the structure of subspaces in a Banach space X. Let a map F : G → S(X) be subadditive, i.e., Assuming that dimensions of all subspaces F (g), for g ∈ G, are finite and do not exceed n, we study conditions that imply that dim g∈G F (g) is finite.
It was proved in [1] that if (*) all F (g) are invariant with respect to a bounded representation of finite multiplicity acting on X then dim g∈G F (g) < Cn for some constant C.
In this talk we consider the situations when one of the conditions (1) or (*) is "satisfied up to a finite-dimensional subspace". We consider complex valued functions on commutative groups with the property that all differences of a certain order belong to a given finite dimensional linear space. If all first order differences of a function belong to a given finite dimensional linear space, then, clearly, the function satisfies a Levi-Civita-type functional equation, hence it is an exponential polynomial. We generalize this observation for higher order differences.

Reference
Patrícia Szokol: Preserving problems related to different means of positive operators (Joint work with Gergő Nagy) In this presentation, we mainly discuss the problem of describing the structure of transformations leaving norms of generalized weighted quasi-arithmetic means of invertible positive operators invariant. Under certain conditions, we present the solution of this problem which generalizes one of our former results containing its solution for weighted quasi-arithmetic means. Moreover, we investigate the relation between the generalized weighted quasi-arithmetic means and the Kubo-Ando means and show that the common members of their class and the set of the latter means are weighted arithmetic means.

Tomasz Szostok: Some remarks on Steffensen's inequality
We present some remarks concerning Steffensen's inequality using the method obtained in [1]. This talk offers a solution to the equality problem of two-variable Bajraktarević means (cf. [2], [3]) with two-variable Cauchy means (cf. [4]), i.e., we aim to solve the functional equation

Reference
where f, g, h, k : I → R are unknown continuous functions such that h, k are differentiable, g, k are nowhere zero on I, f g and h k are strictly monotone on I. For the necessity part of this result, we will assume that f, g : I → R are eight and h, k : I → R are nine times continuously differentiable. Our results extend and generalize those of Alzer and Ruscheweyh ( [1]) who proved that the intersection of the classes of Gini means (that are homogeneous Bajraktarević means) with the class of Stolarsky means (that are homogeneous Cauchy means) is equal to the class of power means (that are homogeneous quasiarithmetic means). Thomas Zürcher: Around a Kazimierz Nikodem result -part II (Joint work with Janusz Morawiec) In the first part, equations of the form were considered. In this talk, we are changing the derivatives f n to some other functions g n , looking for solutions ϕ ∈ L 1 ([0, 1]) of This is not only a cosmetic change. We need new methods to tackle this kind of equations.

Marcin Zygmunt: Remarks on the extensions of multiadditive functions on groups
We present examples of multiadditive functions defined on a subgroup that cannot be extended to the whole group. We continue with a discussion on conditions for which such extensions are possible. This topic is related to [1, Theorem 2].  (Here the norm · is derived from the inner product ·, · .) In the case ε = 0, we simply say that p is nondecreasing. It is very well-known that the gradient of a differentiable convex function is always nondecreasing in the above sense. It is also easy to see that if q : D → R n is nondecreasing and p − q ∞ ≤ ε/2, then p is ε-nondecreasing.

Reference
We have the following Open Problem. Let D ⊆ R n be an open convex set. Does there exist a constant c n ≥ 0 such that, for all ε-nondecreasing function p : D → R n , there exists a nondecreasing function q : D → R n such that p(x) − q(x) ≤ c n ε for all x ∈ D. (1) f (x + y) + g(x + y) + g(x − y) = f (x) + f (y) + 2g(x) + 2g(y).
Putting f 1 := f + g, f 2 := g, f 3 := f + 2g we may write (1) in the form f 1 (x + y) + f 2 (x − y) = f 3 (x) + f 3 (y). Now, using the well known result of L. Székelyhidi, it is easy to show that g = f 2 is a polynomial function of order at most 2 and f = f 1 − f 2 is also a polynomial function of order at most 2. Therefore we have where the functions F 1 and G 1 are additive and G 2 , F 2 are diagonalizations of some 2−additive and symmetric functions. It is also well known that in such case (F 1 , G 1 ) and (F 2 , G 2 ) separately satisfy (1). Thus, using the fact that the functions F 1 and G 1 satisfy (1) and taking x = y in (1), we get i.e., G 1 = 0. Similarly, F 2 (2x) + G 2 (2x) = 2F 2 (x) + 4G 2 (x) yields F 2 = 0. Finally it is visible that c = −2d. This means that f must be additive and g must be quadratic (up to some constants) and the alienation of our equations is proved. This approach is weaker than Ger's result, since stronger assumptions on the domain and codomain of f and g are used here. Such assumptions are needed for the Székelyhidi's results to hold. However it should be noted that this approach is quite universal and may be used to study the alienation problem for a large class of linear functional equations.