The Twenty-fourth Debrecen–Katowice Winter Seminar on Functional Equations and Inequalities Hajdúszoboszló (Hungary), February 6 – 9, 2025

Mihály Bessenyei: Approximation of fractals via Kantorovich iteration (Joint work with Bálint Szabó)

The Kantorovich process provides an alternative method to approximate fractals even in those cases when Hutchinson's approach does not work. In the talk, we present examples for such situations, the implementation of the approximations, and some facts about computational complexity.

Zoltán Boros: Non-zero additive solutions for some alternative equations (Joint work with Rayene Menzer)

Z. Kominek, L. Reich and J. Schwaiger [1] proved, for various particular choices of D ⊆ ℝ², that any additive function f : ℝ → ℝ fulfilling (1) f(x)f(y)=0 for every (x,y)∈D f(x)f(y) = 0\,\,\,\,{\rm{for}}\,{\rm{every}}\,\,\,\,(x,y) \in D has to be equal to zero identically. For instance, their investigations covered the cases when D is the unit circle as well as when D is a one parameter family generated by a pair of ordinary polynomials. 20 years later P. Kutas [2] established the existence of a not identically zero additive function f : ℝ → ℝ fulfilling (1) when D={(x,y)∈ℝ2|xy=1}. D = \{ (x,y) \in {\mathbb R}^2 | xy = 1\}.

In our presentation, we discuss a couple of generalizations of this example. Some of these generalizations are mentioned by P. Kutas in his cited paper as direct applications of his arguments. Another generalization is suggested in the talk as a result of a slightly modified argument.

György Gát: Problems and recent results in the theory of summability of Walsh-Fourier series

Let f^(k):=∫01f(x)ωk(x)dx \hat f(k): = \mathop \smallint \nolimits_0^1 f(x){\omega_k}(x)dx be the kth Walsh-Fourier coefficient of the integrable function f. Let T:=(ti,j)i,j=0∞ T: = ({t_{i,j}})_{i,j = 0}^\infty be a doubly infinite triangular matrix of nonnegative numbers, where ∑k=0n−1tk,n−1=1 \mathop \sum \nolimits_{k = 0}^{n - 1} {t_{k,n - 1}} = 1 for every natural number n. Define the nth matrix transform mean determined by the matrix T of the Fourier series of f as σnT(f):=∑k=0n−1tk,n−1Sk(f). \sigma_n^T(f): = \sum\limits_{k = 0}^{n - 1} {{t_{k,n - 1}}{S_k}(f)}.

We mention some well-known summation methods of this type: —

The Cesàro (or (C, α)) summation,

A much-studied question is whether various summation methods can be used to reconstruct an integrable function. And, if so, in what sense? In this presentation we mention some of the problems and recent achievements [1, 2] in this area. In addition to the Walsh system, we also discuss the trigonometric case.

Attila Gilányi: Bernstein–Doetsch theorem for (M, N)-convex functions (Joint work with Zsolt Páles)

One of the fundamental results of the classical theory of convex functions is the Bernstein–Doetsch theorem ([1]) which states that if a real-valued Jensen-convex function defined on an open interval is locally bounded above at a point in its domain, then it is continuous. In this talk, we present a generalization of this theorem for (M, N)-convex functions, calling a function f : I → J (M, N)-convex (cf., e.g., [3]) if it satisfies the inequality f (M (x, y)) ≤ N (f (x), f (y)) for all x, y ∈ I, where I and J are open intervals, M and N are suitable means on I and J, respectively. Our statement contains Tomasz Zgraja's result on (M, M)-convex functions (cf. [4]) as a special case. It also generalizes the main theorem presented in [2].

Richárd Grünwald: Comparison of Gini means with fixed number of variables (Joint work with Zsolt Páles)

Let us recall the definition of the n-variable Gini mean corresponding to the pair parameters (p, q) ∈ ℝ²: Gp,qnx1,…,xn:=x1p+⋯+xnpx1q+⋯+xnq1p−qif p≠q,expx1plnx1+⋅⋯+xnplnxnx1p+⋯+xnpif p=q, x1,…,xn∈ℝ+. \eqalign{& G_{p,q}^{\left[ n \right]}\left({{x_1}, \ldots,{x_n}} \right): = \left\{{\matrix{{{{\left({{{x_1^p + \cdots + x_n^p} \over {x_1^q + \cdots + x_n^q}}} \right)}^{{1 \over {p - q}}}}} \hfill & {{\rm{if}}\,p \ne q,} \hfill \cr {\exp \left({{{x_1^p\ln \left({{x_1}} \right) + \cdot \cdots + x_n^p\ln \left({{x_n}} \right)} \over {x_1^p + \cdots + x_n^p}}} \right)} \hfill & {{\rm{if}}\,p = q,} \hfill \cr}} \right.\,\, \cr &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left({{x_1}, \ldots,{x_n} \in {{\mathbb R}_+}} \right).\hfill \cr}

Let us consider the global comparison problem of Gini means with fixed number of variables on a subinterval I of ℝ₊, i.e., the following inequality (1) Gr,snx1,…,xn≤Gp,qnx1,…,xn, G_{r,s}^{\left[ n \right]}\left({{x_1}, \ldots,{x_n}} \right) \le G_{p,q}^{\left[ n \right]}\left({{x_1}, \ldots,{x_n}} \right), where n ∈ ℕ, n ≥ 2 is fixed, (p, q), (r, s) ∈ ℝ² and x₁, . . . , x_n ∈ I.

Given a nonempty subinterval I of ℝ₊ and n ∈ ℕ, we introduce the sets Γn(I):={((r,s), (p,s))∈ℝ2×ℝ2| (1) holds for all x1,…,xn∈I},Γ∞(I):=∩n=1∞Γn(I). \matrix{{{\Gamma_n}(I): = \{((r,s),\,(p,s)) \in {{\mathbb R}^2} \times {{\mathbb R}^2}|\,(1)\,{\rm{holds}}\,{\rm{for}}\,{\rm{all}}\,{x_1}, \ldots,{x_n} \in I\},} \hfill \cr {{\Gamma_\infty}(I): = \bigcap\limits_{n = 1}^\infty {{\Gamma_n}(I)}.} \hfill \cr}

In the talk, apart from a limiting case, we characterize the elements of Γ_n(I) via a conditional minimum problem. Then we formulate a conjecture for the aforementioned limiting case.

Eszter Gselmann: A characterization of second-order differential operators (Joint work with Włodzimierz Fechner)

An easy computation shows that if f, g, h ∈ 𝒞²(Ω), then we have (1) d2dx2f⋅g⋅h−fd2dx2g⋅h−gd2dx2f⋅h−hd2dx2f⋅g+f⋅gd2dx2h+f⋅hd2dx2g+g⋅hd2dx2f=0. {{{{\rm{d}}^2}} \over {{\rm{d}}{x^2}}}\left({f \cdot g \cdot h} \right) - f{{{{\rm{d}}^2}} \over {{\rm{d}}{x^2}}}\left({g \cdot h} \right) - g{{{{\rm{d}}^2}} \over {{\rm{d}}{x^2}}}\left({f \cdot h} \right) - h{{{{\rm{d}}^2}} \over {{\rm{d}}{x^2}}}\left({f \cdot g} \right) + f \cdot g{{{{\rm{d}}^2}} \over {{\rm{d}}{x^2}}}h + f \cdot h{{{{\rm{d}}^2}} \over {{\rm{d}}{x^2}}}g + g \cdot h{{{{\rm{d}}^2}} \over {{\rm{d}}{x^2}}}f = 0.

As a generalization of the Leibniz rule, usually the identity d2dx2f⋅g=d2dx2f⋅g+2ddxf⋅ddxg+f⋅d2dx2g(f,g∈C2(Ω)) \matrix{{{{{{\rm{d}}^2}} \over {{\rm{d}}{x^2}}}\left({f \cdot g} \right) = {{{{\rm{d}}^2}} \over {{\rm{d}}{x^2}}}f \cdot g + 2{{\rm{d}} \over {{\rm{d}}x}}f \cdot {{\rm{d}} \over {{\rm{d}}x}}g + f \cdot {{{{\rm{d}}^2}} \over {{\rm{d}}{x^2}}}g} & {(f,g \in {{\cal C}^2}(\Omega))} \cr} is considered. In some sense, the disadvantage of the latter identity, compared to identity (1), is that it includes not only the second-order, but also the first-order differential operator.

Let k be a fixed nonnegative integer and Ω ⊂ ℝ be a nonempty and open set. In this talk, we study operators D : 𝒞^k(Ω) → 𝒞(Ω) that fulfil (2) D(f⋅g⋅h)−fD(g⋅h)−gD(f⋅h)−hD(f⋅g)+f⋅gD(h)+f⋅hD(g)+g⋅hD(f)=0 D(f \cdot g \cdot h) - fD(g \cdot h) - gD(f \cdot h) - hD(f \cdot g) + f \cdot gD(h) + f \cdot hD(g) + g \cdot hD(f) = 0 for all f, g, h ∈ 𝒞^k(Ω). We emphasize that unless stated otherwise, the operator D is not assumed to be linear. As we can observe, (2) turns out to be suitable for characterizing second-order linear differential operators in function spaces.

Mehak Iqbal: A functional equation for monomial functions (Joint work with Eszter Gselmann)

Let 𝕂 be a field of characteristic zero, 𝔽 ⊂ 𝕂 be a subfield, n be a positive integer and κ be in 𝕂. Let further f : 𝔽 → 𝕂 be a generalized monomial of degree n. In this talk, we focus on the functional equation f(x2)=κ⋅xnf(x) (x∈𝔽) \matrix{{f({x^2}) = \kappa \cdot {x^n}f(x)\;} & {(x \in {\mathbb F})} \cr} for the unknown generalized monomial f. The presented results extend the classical cases n = 1, 2, that is, the cases of additive and quadratic functions, respectively.

Tibor Kiss: On continuously differentiable solutions of a functional equation (Joint work with Péter Tóth)

The general solution of the composite functional equation Fx+y2+f1x+f2y=Gg1x+g2y,x,y∈I, \matrix{{F\left({{{x + y} \over 2}} \right) + {f_1}\left(x \right) + {f_2}\left(y \right) = G\left({{g_1}\left(x \right) + {g_2}\left(y \right)} \right),} & {x,y \in I,} \cr} was known if the functions involved are continuously differentiable and the derivatives of g₁ and g₂ are nowhere zero. In the talk we are going to demonstrate how to eliminate this latter condition and solve the equation in question under continuous differentiability assumption.

Judit Makó: Approximate Jensen convexity

In the talk, some new results concerning approximate Jensen convexity are presented. Let X be a normed space, D ⊂ X be a convex set, and c > 0. A function f : D → ℝ is (c, α)-Jensen convex, if for all x, y ∈ D fx+y2≤cf(x)+cf(y)+α(x−y). f\left({{{x + y} \over 2}} \right) \le cf(x) + cf(y) + \alpha (\left\| {x - y} \right\|).

We investigate the (c, α)-Jensen convex functions in different concepts and answer questions: How can we strengthen this inequality? Is the above inequality superstable? How from a Hermite–Hadamard type inequality can we get a (c, α)-Jensen convex function? We also obtain Bernstein–Doetsch type results.

Magdalena Mamcarz: Inequalities for generalized convex functions with the use of the Hesselager theorem

We observe that the theorem of Hesselager [2] may be used to obtain new proofs of Hermite-Hadamard inequalities for functions that are convex with respect to a Chebyshev system. Such inequalities were proved by Bessenyei and Páles [1] for integrals with density function. In our approach, the first function from the system involved must be constant but the main theorem is valid for the integral with respect to any measure. Additionally, we present examples of inequalities that follow from Hesselager's theorem but are not of the Hermite–Hadamard type.

Rayene Menzer: An alternative equation for polynomial functions on hyperbolas (Joint work with Zoltán Boros)

In our presentation, we establish the following result:

This research is motivated by analogous investigations in [4] for one additive function f fulfilling f (x)f (y) = 0 under such constraints, by our previous results in [2] and [3], when we considered one generalized monomial f or a generalized polynomial of degree two, for some rational number m, as well as by similar investigations by Z. Boros, W. Fechner in [1] for one real generalized polynomial in the particular case m = −1 (when our algebraic constraint describes the unit circle).

Gergő Nagy: On the σ-balancing property of generalized quasi-arithmetic means of n variables (Joint work with Tibor Kiss)

We say that a mean M : I² → ℝ is balanced if M(M(x,Mx,y MM(x,y),y=M(x,y) M(M(x,M\left({x,y\;M\left({M(x,y),y} \right)} \right) = M(x,y) holds for all x, y ∈ I. In the 1930s, Aumann proved that the balancing property characterizes quasi-arithmetic means among analytic or Cauchy means. In the talk, we are going to extend the above concept to the n > 2 variable case, and investigate it in the class of generalized quasi-arithmetic means. Our related result is analogous to Aumann's statement.

Andrzej Olbryś: On some version of separation theorem

Let X be an abelian group, let f, g : X → ℝ, g ≤ f and ω : X × X → ℝ be given functions satisfying condition: af(x,y)≤ω(x,y)≤ag(x,y),x,y∈X, \matrix{{{a_f}(x,y) \le \omega (x,y) \le {a_g}(x,y),} & {x,y \in X,} \cr} where a_k : X ×X → ℝ stands for a Cauchy's difference of the function k : X → ℝ, i.e. akx,y:=kx+y−kx−ky. {a_k}\left({x,y} \right): = k\left({x + y} \right) - k\left(x \right) - k\left(y \right).

In our talk, we discuss sufficient conditions that guarantee the existence of a function h : X → ℝ such that g(x) ≤ h(x) ≤ f (x), x ∈ X, and ωx,y=ahx,y,x,y∈X. \matrix{{\omega \left({x,y} \right) = {a_h}\left({x,y} \right),} & {x,y \in X.} \cr}

Zsolt Páles: On the equality of generalized Bajraktarević means under first-order differentiability assumptions (Joint work with Amr Zakaria)

In the talk, we consider the equality problem of generalized Bajraktarević means, i.e., we are going to solve the functional equation (*) f−1p1x1fx1+⋅.⋅.⋅.+pnxnfxnp1x1++pnxn=g−1q1x1gx1+⋅.⋅.⋅.+qnxngxnq1x1++qnxn, (*)\,{f^{- 1}}\left({{{{p_1}\left({{x_1}} \right)f\left({{x_1}} \right) + \cdot. \cdot. \cdot. + {p_n}\left({{x_n}} \right)f\left({{x_n}} \right)} \over {{p_1}\left({{x_1}} \right) + + {p_n}\left({{x_n}} \right)}}} \right) = {g^{- 1}}\left({{{{q_1}\left({{x_1}} \right)g\left({{x_1}} \right) + \cdot. \cdot. \cdot. + {q_n}\left({{x_n}} \right)g\left({{x_n}} \right)} \over {{q_1}\left({{x_1}} \right) + + {q_n}\left({{x_n}} \right)}}} \right), which holds for all x = (x₁, . . . , x_n) ∈ Iⁿ, where n ≥ 2, I is a nonempty open real interval, the unknown functions f, g : I → ℝ are strictly monotone, and the vector-valued weight functions p=(p1,…,pn):I→ℝ+n p = ({p_1}, \ldots,{p_n}):I \to {\mathbb R}_+^n and q=(q1,…,qn):I→ℝ+n q = ({q_1}, \ldots,{q_n}):I \to {\mathbb R}_+^n are also unknown. This equality problem in the symmetric two-variable case (i.e., when n = 2 and p₁ = p₂, q₁ = q₂) was solved under sixth-order regularity assumptions by Losonczi [2]. In [3] the same conclusion was established assuming only first-order differentiability. In [1] the nonsymmetric case was considered. Assuming third-order differentiability of f, g and the first-order differentiability of at least three of the functions p₁, . . . , p_n, it was proved that (*) holds if and only if there exist four constants a, b, c, d ∈ ℝ with ad ≠ bc such that cf+d>0,g=af+bcf+d,andqℓ=(cf+d)pℓ(ℓ∈{1,…,n}). \matrix{{cf + d > 0,} \hfill & {g = {{af + b} \over {cf + d}},} \hfill & {{\rm{and}}} \hfill & {q\ell = (cf + d)p\ell} \hfill & {(\ell \in \{1, \ldots,n\}).} \hfill \cr}

The goal is to verify the same conclusion under first-order differentiability conditions.

Patryk Rela: The Orlicz risk measure under uncertainty

Let (Ω, ℱ) is a measurable space and μ : ℱ → [0, 1] is a capacity, that is a monotone set function, satisfying μ(∅) = 0 and μ(Ω) = 1. The risk is represented by the ℱ-measurable function X : Ω → [0, ∞) such that μ({X > t}) = 0 for some t ∈ ℝ. The Orlicz risk measure under uncertainty for the risk X is defined as the solution π_(μ,α,Φ)[X, x] of the equation (1) EμΦ(X−x)+πμ,α,ΦX,x−x=1−α, {E_\mu}\left[ {\Phi \left({{{{{(X - x)}_ +}} \over {{\pi_{\left({\mu,\alpha,\Phi} \right)}}\left[ {X,x} \right] - x}}} \right)} \right] = 1 - \alpha, where α ∈ [0, 1) is a given parameter, x ∈ ℝ is fixed real number and Φ: [0, ∞) → [0, ∞) is a normalized Young function, that is a strictly increasing, convex function Φ: [0, ∞) → [0, ∞) satisfying Φ(0) = 0, Φ(1) = 1 and lim_z→∞ Φ(z) = ∞. E_μ is the Choquet integral with respect to the capacity μ, defined by EμX:=∫0∞μ({X>k})dk. {E_\mu}\left[ X \right]: = \mathop \smallint \nolimits_0^\infty \mu (\{X > k\})dk.

The aim of this talk is to prove the existence and uniquenes of the Orlicz risk measure defined by (1) and to characterize its several important properties.

Maciej Sablik: Portfolio selection based on a fuzzy measure (Joint work with Timothy Nadhomi and Chisom P. Okeke)

Effective portfolio risk management is a critical component for both investors and fund managers to achieve their financial objectives while navigating asset selection. This paper explores the application of non-additive fuzzy measures as an alternative to traditional risk metrics like standard deviation. We briefly discuss the limitations of Modern Portfolio Theory (MPT) and its reliance on normal distribution assumptions. To address these limitations, we introduce non-additive fuzzy measures, which do not assume specific probability distributions. This approach accommodates imprecision and uncertainty in financial markets, providing a more comprehensive understanding of portfolio risk. By considering diversification and asset characteristic dependencies, non-additive fuzzy measures offer a promising avenue for more accurate risk analysis and informed investment decisions.

László Székelyhidi: On the powers of maximal ideals in Fourier algebras

The Fourier algebra of a locally compact abelian group plays a basic role in abstract harmonic analysis. In particular, the maximal ideals of this algebra are the corner stones of spectral analysis and synthesis. In this work we present some useful descriptions of the powers of maximal ideals in the Fourier algebra of ℝⁿ. Our descriptions are related to the Weierstrass Preparation Theorem. One possible form of this theorem can be formulated in the following way:

Let n be a positive integer and m a nonnegative integer. Let z₀ be in ℂ, further let p₀ = (p₀₁, p₀₂, . . . , p_0n) be in ℂⁿ. Assume that f is a complex valued analytic function in a neighborhood of (z₀, p₀), and z₀ is an m-multiple root of the equation f (z, p₀) = 0, that is, ∂1kfz0,p0=0,k=0,1,…,m−1,∂1mfz0,p0≠0. \matrix{{\partial_1^kf\left({{z_0},{p_0}} \right) = 0,} & {k = 0,1, \ldots,m - 1,} & {\partial_1^mf\left({{z_0},{p_0}} \right) \ne 0.} \cr}

Then there exists a neighborhood of (z₀, p₀) in which f can be expressed in the form f(z,p)=[(z−z0)m+am−1(p)(z−z0)m−1+am−2(p)(z−z0)m−2+⋯+a0(p)]b(z,p), f(z,p) = [{(z - {z_0})^m} + {a_{m - 1}}(p){(z - {z_0})^{m - 1}} + {a_{m - 2}}(p){(z - {z_0})^{m - 2}} + \cdots + {a_0}(p)]b(z,p), where a_i(p₀) = 0, the functions a_i, b (i = 0, 1, . . . , m − 1) are analytic and uniquely determined by f, further b(z₀, p₀) ≠ 0.

Tomasz Szostok: Inequalities involving the Choquet integral (Joint work with Jacek Chudziak)

For an increasing function f, we write the Choquet integral (C) ∫abfxdμx=∫0∞μs∈a, b : fs≥xdx (C)\,\mathop \smallint \nolimits_a^b f\left(x \right)d\mu \left(x \right) = \mathop \smallint \nolimits_0^\infty \mu \left({\left\{{s \in \left[ {a,\;b} \right]\;:\;f\left(s \right) \ge x} \right\}} \right)dx (μ is here a capacity) in the form of a Stieltjes integral. Then, using this representation, we study some basic inequalities (like Hermite–Hadamard, Jensen or Ostrowski).

Norbert Tóth: Testing non-solvability of linear programs (Joint work with Mihály Bessenyei)

It may occur that only the solvability issue of a linear program counts in a practical application (and the solution itself, if it exists, does not). Motivated by this phenomena, we suggest an implementable method for testing nonsolvability of linear programs.

Péter Tóth: Generalized inverses of strictly monotone transformations of the plane

Our talk concerns a problem proposed by Z. Páles during the 60th International Symposium on Functional Equations [2]. Let n ∈ ℕ and K ⊆ ℝⁿ be a nonempty, convex set. Suppose that the mapping f : K → ℝⁿ is strictly monotone, i.e., it fulfills the inequality 〈fx−fy,x−y〉>0 \langle f\left(x \right) - f\left(y \right),x - y\rangle > 0 for all x, y ∈ K such that x ≠ y, where 〈·, ·〉 denotes the standard inner product on the Euclidean space ℝⁿ. Observe that for n = 1 this is indeed equivalent to the strict monotonicity of f on an interval.

The question is: does f have a generalized left inverse function defined on conv (f (K)) which is monotone? More precisely, does there exist a function f ⁽⁻¹⁾ : conv (f (K)) → K such that f⁽⁻¹⁾ (f (x)) = x for all x ∈ K, moreover f−1(u)−f−1(v), u−v≥0 \left\langle {{f^{\left({- 1} \right)}}(u) - {f^{\left({- 1} \right)}}(v),\,u - v} \right\rangle \ge 0 for all u, v ∈ conv (f (K))?

It is known (see [1]) that for n = 1 such a function f⁽⁻¹⁾ exists and it is unique. In our talk we consider the case n = 2. We are going to demonstrate that this generalized inverse does not exist for an arbitrary convex domain K ⊆ ℝ². However, we show that if K is closed, convex then f⁽⁻¹⁾ exists and it is uniquely determined. Our proof is based on a classical convex geometric theorem of Helly.